Convergence analysis of explicit stabilized integrators for parabolic semilinear stochastic PDEs
aa r X i v : . [ m a t h . NA ] F e b Convergence analysis of explicit stabilized integrators forparabolic semilinear stochastic PDEs
Assyr Abdulle , Charles-Edouard Br´ehier , and Gilles Vilmart February 8, 2021
Abstract
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicitmethods to avoid the severe timestep restriction faced by standard explicit integratorsapplied to stiff diffusion problems. In this paper, we provide a fully discrete strongconvergence analysis of a family of explicit stabilized methods coupled with finiteelement methods for a class of parabolic semilinear deterministic and stochastic partialdifferential equations. Numerical experiments including the semilinear stochastic heatequation with space-time white noise confirm the theoretical findings.
Keywords: explicit stabilized methods, second kind Chebyshev polynomials, stochas-tic partial differential equations, finite element methods.
AMS subject classification (2020):
In this paper, we consider semilinear parabolic stochastic partial differential equations(SPDEs), in the framework of [11], of the form du ( t ) = Λ u ( t ) dt + F ( u ( t )) dt + σdW Q ( t ) , u (0) = u , (1)where u is a given initial condition, Λ is a symmetric diffusion operator, and F is a smoothnonlinearity, σ > (cid:0) W Q ( t ) (cid:1) t ≥ is a Q -Wiener process. The semi-discrete approximationobtained by spatial discretization using the finite element method with piecewise linearelements is defined by the following equation: for a spatial mesh size h ∈ (0 , du h ( t ) = Λ h u h ( t ) dt + P h F ( u h ( t )) dt + σP h dW Q ( t ) , u h (0) = P h u , (2)where P h denotes the L -orthogonal projection operator onto the finite element space. Thestochastic evolution problem (2) is driven by a Q h -Wiener process, with Q h = P h QP h .The spatial discretization Λ h of the diffusion operator Λ typically yields large eigenval-ues, with spectral radius of size O ( h − ) for a Laplace operator and a finite element meshwith size h , which yields a severe timestep restriction τ = O ( h ) ≪ Mathematics Section, ´Ecole Polytechnique F´ed´erale de Lausanne, Station 8, 1015 Lausanne, Switzer-land, Assyr.Abdulle@epfl.ch Univ Lyon, CNRS, Universit´e Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, F-69622Villeurbanne, France. [email protected] Universit´e de Gen`eve, Section de math´ematiques, 2-4 rue du Li`evre, CP 64, 1211 Gen`eve 4, Switzerland,[email protected] n ≥ u hn +1 = ( I − τ Λ h ) − (cid:0) u hn + P h τ F ( u hn ) + σ ∆ W Qn (cid:1) , (3)where h ∈ (0 ,
1] and τ ∈ (0 , W Qn = W Q ( t n +1 ) − W Q ( t n ) with t n = nτ . Alternatively to implicit methods, in this paper weconsider families of explicit stabilized schemes of the form u hn +1 = A s ( τ Λ h ) u hn + B s ( τ Λ h ) P h (cid:0) τ F ( u hn ) + σ ∆ W Qn (cid:1) , (4)where A s and B s are polynomials of degree s = s ( τ, h ) which are chosen to satisfy a suit-able stability condition depending on τ and h . For well-chosen polynomials, the stabilitydomain can be very large, which allow to choose the time-step size τ independently of themesh size h . In particular, we shall consider variants of the SK-ROCK method from [2].Explicit stabilized integrators are well known efficient time integrators for stiff prob-lems, in particular arising from diffusion PDE problems, both in the deterministic andstochastic settings, see [13, Sect. IV.2] and the review [1]. Compared to the explicit sta-bilized integrators first proposed in the stochastic context in [3, 4] using only first kindChebyshev polynomials and a large damping parameter η , a new optimal family of explicitstabilized method involving second kind Chebyshev polynomials, analogous to (4), was in-troduced in [2]. It permits the efficient integration of stiff systems of stochastic differentialequations, with favourable mean-square stability properties and high-order for samplingthe invariant measure of ergodic problems such as the overdamped Langevin equation.While the convergence analysis of stochastic explicit stabilized methods is known infinite dimension, see e.g. [3, 4, 6, 7], applied to a semi-discrete in space problem (2),in general one obtains convergence estimates with error constants that depend on thespatial mesh size h . The aim of this paper is to remove this dimension dependency and toprovide the first strong convergence analysis of families of explicit stabilized methods inthe context of semilinear parabolic SPDEs such as the stochastic heat equation, with errorconstants that are independent of the space and time mesh parameters h and τ . Undernatural assumptions, for a fixed time interval size T >
0, we prove that the families ofexplicit stabilized schemes (4) satisfy the following space-time strong error estimate. Forall α ∈ [0 ,
1] chosen small enough such that ( − Λ) ( α − / Q / is a bounded Hilbert-Schmidtoperator (corresponding to α ∈ [0 , /
2) for the one-dimensional stochastic heat equationwith space time-white noise), we show that the mean-square strong error satisfies (cid:0) E | u ( t n ) − u hn | (cid:1) ≤ C (1 + | u | t − α n )( τ α + h α ) , for all τ, h ∈ (0 ,
1] with t n = nτ ≤ T , where C is independent of n and both the space andtime mesh parameters h, τ and the degree s = s ( τ, h ). The constant C depends howeveron T and α . The parameter α is related to the spatial and temporal regularity of theprocess. Note that in the deterministic case ( σ = 0 in (1)), we obtain an order one ofconvergence in time and order two in space, corresponding formally to α →
2. Here, | · | denotes the L ( D ) norm in space on a bounded, open and convex polyhedral domain D in dimension d .While the scheme (4) achieves the same strong convergence rate as the classical linearimplicit Euler method (3), as first studied in [19], we emphasize that the analysis in this2aper is not a straightforward generalization of that in [19] due to lower regularizationproperties of the explicit numerical flow for the diffusion A s ( τ Λ h ) in (4) compared to thelinear implicit Euler method (3). Furthermore, the convergence analysis of an explicitmethod for parabolic SPDEs with space and time mesh independent constants is a newresult, to the best of our knowledge. In addition, note that the convergence analysis is per-formed for a general class of explicit stabilized methods satisfying suitable regularizationconditions, and it provides a unified natural framework for Runge-Kutta type schemes,including the linear implicit Euler method (3), applied to semilinear parabolic SPDEs.This paper is organized as follows. In Section 2 we describe the classical Hilbert spacesetting and assumptions used for the analysis of SPDEs and of their numerical approxi-mations. Section 3 is devoted to the definition of the explicit stabilized method in time,coupled with a standard finite element method in space and presents our main convergenceresults. Section 4 is dedicated to the strong convergence analysis of the method. FinallySection 5 is dedicated to numerical experiments which confirm the theoretical findings. Let H be the separable Hilbert space L ( D ), where D ⊂ R d is an open and convexpolyhedral bounded domain in dimension d ∈ { , , } . The inner product in H is denotedby h· , ·i . The norm in H is denoted by | · | . The operator norm on the space L ( H ) ofbounded linear operators from H to H is denoted by k · k L ( H ) . The Hilbert-Schmidt normon the space L ( H ) of Hilbert-Schmidt operators on H is denoted by k · k L ( H ) .The assumptions stated in this section are standard in the literature on SPDEs [11]and their numerical approximations [16], [18]. Λ The linear operator Λ is defined as the linear second-order elliptic differential operatorΛ u ( x ) = div (cid:0) a ( x ) ∇ u ( x ) (cid:1) , with domain D (Λ) = H ( D ) ∩ H ( D ), where we consider for simplicity homogeneousDirichlet boundary conditions on the boundary of the domain ( u = 0 on ∂ D ). We assumethat the field a : D → R is continuous on the closure D of D , of class C ∞ on D , andsatisfies min x ∈D a ( x ) >
0. The following result is standard.
Proposition 2.1.
The linear operator Λ is unbounded and self-adjoint. There exists acomplete orthonormal system (cid:0) e m (cid:1) m ≥ of H , and a non-decreasing sequence (cid:0) λ m (cid:1) m ≥ ,such that one has λ > , lim m →∞ m − d λ m ∈ (0 , ∞ ) , and for all m ≥ , Λ e m = − λ m e m . The linear operator Λ generates a strongly continuous semi-group on H , denoted by (cid:0) e t Λ (cid:1) t ≥ . For all u ∈ H and all t ≥
0, one has e t Λ u = X m ≥ e − λ m t h u, e m i e m .
3n addition, for any α ∈ [ − , − Λ) α is defined as follows: for all u ∈ H , one has ( − Λ) α u = X m ≥ λ αm h u, e m i e m . The semi-group (cid:0) e t Λ (cid:1) t ≥ satisfies the following smoothing and regularity properties: forany γ ∈ [0 , t ≥ t γ (cid:13)(cid:13) ( − Λ) γ e t Λ (cid:13)(cid:13) L ( H ) < ∞ , sup t> t − γ k (cid:0) e t Λ − I )( − Λ) − γ k L ( H ) < ∞ . (5) F The nonlinear operator F in the stochastic evolution equation (1) is assumed to be globallyLipschitz continuous. Assumption 2.2.
The nonlinear operator F is a globally Lipschitz continuous mappingfrom H to H . Precisely there exists a constant C such that for all u, v ∈ H , | F ( u ) − F ( v ) | ≤ C | u − v | . For instance, let f : R → R be a globally Lipschitz continuous real-valued function. If F is defined as the associated so-called Nemytskii operator, precisely F ( u ) = f ( u ( · )) forall u ∈ H , then Assumption 2.2 is satisfied. The local Lipschitz continuous case is outof the scope of this article, and would require to modify the scheme (using for instanceimplicit methods, splitting [8] or tamed or truncated explicit [14] schemes, adaptive time-stepping [15] techniques). Q -Wiener process and mild solution of the SPDE (1) Let us first introduce the Q -Wiener process (cid:0) W Q ( t ) (cid:1) t ≥ . Let (cid:0) ǫ m (cid:1) m ≥ be a completeorthonormal system of the Hilbert space H , and consider a bounded sequence (cid:0) q m (cid:1) m ≥ ofnonnegative real numbers. Let Q and Q be the bounded, linear, self-adjoint operatorson H , defined by Qu = X m ≥ q m h u, e m i e m , Q u = X m ≥ √ q m h u, e m i e m . Definition 2.3.
Let (cid:0) β m (cid:1) m ≥ be a family of independent standard real-valued Wienerprocesses, β m = (cid:0) β m ( t ) (cid:1) t ∈ R + , defined on a probability space which satisfies the usual con-ditions. The Q -Wiener process (cid:0) W Q ( t ) (cid:1) t ∈ R + is defined as follows: for all t ≥ , W Q ( t ) = X m ≥ √ q m β m ( t ) ǫ m . Note that the Q -Wiener process W Q takes values in H if and only if P m ≥ q m < ∞ (which means that Q is a trace-class operator, or equivalently that Q is an Hilbert-Schmidt operator). Using the regularization properties (5) for the semi-group (cid:0) e t Λ (cid:1) t ≥ generated by Λ, it is possible to define solutions of stochastic evolution equations driven4y a Q -Wiener process without requiring that Q is trace-class. More precisely, the well-posedness of the stochastic evolution equation (1) is ensured by the assumption that theparameter α defined in Assumption 2.4 below is positive. We refer for details to [11,Chap. 5] in the context of linear SPDEs with additive noise and to [11, Sect. 7.1] in thecontext of semilinear SPDEs. Assumption 2.4.
Assume that there exists α > such that k ( − Λ) α − Q k L ( H ) < ∞ ,and define α = sup n α ∈ [0 , k ( − Λ) α − Q k L ( H ) < ∞ o . In the sequel, the parameter α plays a key role, indeed it determines the spatial andtemporal regularity properties of the solutions, as well as strong rates of convergence of thenumerical methods. We observe that α ∈ (0 ,
1] and for all α ∈ [0 , α ), k ( − Λ) α − Q k L ( H ) < ∞ . We recall that for an equation driven by space-time white noise in dimension d = 1( Q = I ), one has α = . However, if d ≥ Q = I , then k ( − Λ) α − k L ( H ) = P m ≥ λ α − m = ∞ for all α ≥ Q = I ) if d ≥ u ∈ H and any time T ∈ (0 , ∞ ), the stochastic evolution equation (1) admits a unique global mild solution, whichsatisfies u ( t ) = e tA u + Z t e ( t − s )Λ F ( u ( s )) ds + σ Z t e ( t − s )Λ dW Q ( s ) , for all t ∈ [0 , T ] , (6)see [17, Chap. 2, Thm. 2.25], and see Proposition A.1 (Appendix) for more details andtemporal and spatial regularity estimates. The aim of this section is to define the semi-discrete and full-discrete approximations ofthe process (cid:0) u ( t ) (cid:1) t ≥ . We first describe the spatial discretization, performed using a finiteelement method. We then detail the fully-discrete scheme, which is the main topic of thisarticle: the temporal discretization is performed using explicit-stabilized integrators. Weprovide several concrete examples of such schemes using Chebyshev polynomials, and statea general assumption for the analysis. Finally, we state our main convergence results. For every mesh size h > h ∈ (0 , T h be atriangulation of the bounded, convex, polyhedral domain D . Assume that the triangulationis quasi-uniform. The finite element spaces V h are finite dimensional linear subspaces of V = H ( D ). Let N h denote the dimension of V h . For simplicity, we consider V h to bethe space of continuous functions on D , which are piecewise linear on T h and zero at theboundary ∂ D .For all h ∈ (0 , h on V h , equipped with the inner prod-uct h· , ·i inherited from H = L ( D ), as follows: for all u h , v h ∈ V h , h Λ h u h , v h i = − Z D a ( x ) ∇ u h ( x ) · ∇ v h ( x ) dx. h is a negative and self-adjoint linear operator on V h , thus there exists anorthonormal basis (cid:0) e m,h (cid:1) ≤ m ≤ N h of V h and positive real numbers (cid:0) λ m,h (cid:1) ≤ m ≤ N h such thatΛ h e m,h = − λ m,h e m,h , ≤ m ≤ N h , and λ ,h ≤ · · · ≤ λ m,h ≤ λ m +1 ,h ≤ · · · ≤ λ N h ,h . In addition, for all h >
0, the spectral gap λ ,h = inf u h ∈ V h h− Λ h u h ,u h i| u h | of − Λ h is larger or equal than the spectral gap λ = inf u ∈ H h− Λ u,u i| u | of − Λ (this is an immediate consequence of the inclusion V h ⊂ H ).Define P h : H → V h as the L -orthogonal projection operator onto V h (for the innerproduct h· , ·i ), for all h ∈ (0 , u ∈ H , one has P h u = N h X m =1 h u, e m,h i e m,h . In the sequel, the following notation is extensively used. For any function φ : ( −∞ , → R , and all h ∈ (0 , φ (Λ h ) as follows: for all u h ∈ V h , φ (Λ h ) u h = N h X m =1 φ ( − λ m,h ) h u h , e m,h i e m,h . (7)In particular, the linear operators ( − Λ h ) α , for all α ∈ [ − , e t Λ h , for all t ≥
0, aredefined by the expression (7) with φ ( z ) = ( − z ) α and φ ( z ) = e tz respectively.Let us state the conditions required for the analysis below. Those conditions aresatisfied for piecewise linear finite element methods on a quasi-uniform triangulation ofthe bounded, convex, polygonal domain D . We refer for instance to [17, Section 3.2] andreferences therein for details. Assumption 3.1.
For all α ∈ [ − , ] , there exists C α ∈ (0 , ∞ ) such that for all h ∈ (0 , and all u h ∈ V h , C − α | ( − Λ h ) α u h | ≤ | ( − Λ) α u h | ≤ C α | ( − Λ h ) α u h | , and for all u ∈ D (cid:0) ( − Λ) α (cid:1) , | ( − Λ h ) α P h u | ≤ C α | ( − Λ) α u | . Moreover, for every α ∈ [0 , , α ∈ [ α , , there exists C α ,α ∈ (0 , ∞ ) such that for all h ∈ (0 , , (cid:13)(cid:13) ( − Λ) α (cid:0) I − P h (cid:1) ( − Λ) − α (cid:13)(cid:13) L ( H ) ≤ C α ,α h α − α . It is then straightforward to prove the following result.
Proposition 3.2.
Let Assumptions 2.4 and 3.1 be satisfied. Then for all α ∈ [0 , α ) , sup h ∈ (0 , k ( − Λ h ) α − P h Q k L ( H ) < ∞ . The semi-discrete approximation of (1) obtained by spatial discretization using thefinite element method is defined by the following equation: for h ∈ (0 , du h ( t ) = Λ h u h ( t ) dt + P h F ( u h ( t )) dt + σP h dW Q ( t ) , u h (0) = P h u . (8)6his stochastic evolution equation is driven by a Q h -Wiener process, with Q h = P h QP h .The initial condition is given by u h = P h u . The problem is globally well-posed, and itsunique solution (cid:0) u h ( t ) (cid:1) t ≥ takes values in the finite dimensional space V h . It satisfies thefollowing mild formulation u h ( t ) = e t Λ h u h + Z t e ( t − s )Λ h P h F ( u h ( s )) ds + σ Z t e ( t − s )Λ h P h dW Q ( s ) , (9)where we recall that the semi-group is defined by e t Λ h u h = P N h m =1 e − tλ m,h h u h , e m,h i e m,h for all t ≥ u h ∈ V h using (7). The following smoothing and regularity properties(which are uniform in h ∈ (0 , γ ∈ [0 , h ∈ (0 , sup t ≥ t γ (cid:13)(cid:13) ( − Λ h ) γ e t Λ h (cid:13)(cid:13) L ( V h ) < ∞ . (10)and sup h ∈ (0 , sup t> t − γ k (cid:0) e t Λ h − I )( − Λ h ) − γ k L ( V h ) < ∞ , (11)where k · k L ( V h ) denotes the operator norm on L ( V h ). The objective of this section is to describe the temporal discretization of (8) using explicit-stabilized integrators. More precisely, we consider families of functions (cid:0) A s (cid:1) s ≥ and (cid:0) B s (cid:1) s ≥ , indexed by the integer parameter s ≥ τ > τ ∈ (0 , u hn +1 = A s ( τ Λ h ) u hn + B s ( τ Λ h ) P h (cid:0) τ F ( u hn ) + σ ∆ W Qn (cid:1) , (12)for all n ≥
0, with the initial condition u h = P h u = u h (0). In (12), the Wiener incrementsare defined by ∆ W Qn = W Q ( t n +1 ) − W Q ( t n ), with t n = nτ . The self-adjoint linearoperators A s ( τ Λ h ) and B s ( τ Λ h ) are defined using (7): A s ( τ Λ h ) = N h X m =1 A s ( − τ λ m,h ) h· , e m,h i e m,h , B s ( τ Λ h ) = N h X m =1 B s ( − τ λ m,h ) h· , e m,h i e m,h . (13)From (12), one gets a discrete-time mild formulation, similar to (6) and (9): for all n ≥ u hn = A s ( τ Λ h ) n u h + n − X k =0 A s ( τ Λ h ) n − − k B s ( τ Λ h ) P h (cid:0) τ F ( u hk ) + σ ∆ W Qk (cid:1) . (14)For given time-step size τ and the mesh size h , the parameter s is chosen large enoughsuch that the following stability condition is satisfied: τ λ N h ,h ≤ L s , (15)where the size L s ∈ (0 , ∞ ] of the stability domain depends on the choice of A s . One ofthe minimal requirements (see Assumption 3.3 below for the full list of conditions) is thatsup z ∈ [ − L s , |A s ( z ) | ≤ . (16)7 - - - -
20 0 - - - - - -
20 0
Figure 1: Stability functions A s ( z ) and B s ( z ) in (19) of the explicit stabilized methods(normal and logarithmic scales), with degree s = 7, damping parameter η = 0 . A s ( z ) is a (non-constant) polynomial, then |A s ( z ) | → ∞ as | z | → ∞ , whichyields that L s < ∞ necessarily has a finite value, whereas L s = + ∞ for the rationalfunction A ( z ) = (1 − z ) − related to the implicit Euler method. Several choices are possiblefor the definition of (cid:0) A s (cid:1) s ≥ and (cid:0) B s (cid:1) s ≥ and will be discussed in the next section. Let us provide examples of explicit stabilized integrators inspired from the literature andwhich fit in this framework, and discuss implementation details. The first choice is inspiredfrom [2] in the context of stiff stochastic integrators with favourable mean-square stabilityproperties, while the second choice in inspired from [12] in the context of deterministicpartitioned stabilized Runge-Kutta methods for convex optimisation.The idea of explicit stabilized methods is to consider suitably chosen polynomials A s such that the stability domain size L s in (16) grows rapidly with s . Under the consistencycondition A s ( z ) = 1 + z + O ( z ) ( z → L s is given by A s = T s (1 + z/s ), where T s is the firstkind Chebyshev polynomial of order s satisfying T s (cos θ ) = cos( sθ ), giving L s = 2 s . Thisquadratic growth of L s with respect to s permits to achieve much larger stability domains,in contrast to standard explicit Runge-Kutta methods. However, it turns out, both in thedeterministic and stochastic settings [1], that the condition (16) is not sufficient in generalfor the design of a robust integrator, and some damping should be introduced to obtain astronger bound sup z ∈ [ − L s , − δ ] | A s ( z ) | ≤ − ǫ, where ǫ > δ > s and the mesh sizes τ, h . A8atural choice, as considered in this article, is to consider a fixed damping parameter η > A s ( z ) as A s ( z ) = T s ( ω + ω z ) T s ( ω ) , (17)where T s is the first-kind Chebyshev polynomial of degree s , and the parameters ω , ω are defined by ω = 1 + ηs , ω = T s ( ω ) T ′ s ( ω ) . (18)where η is referred to as the damping parameter. The real numbers ω and ω depend on s and η , but the dependence is omitted to simplify notation.Definition (17)-(18) yields a stability domain with length L s ≥ ω in (16), and it can be shown that L s ≥ (2 − η ) s , with a small and fixed dampingparameter η >
0. Observe that L s grows quadratically as a function of s , and is arbitrarilyclose to the optimal value L s = 2 s , see [1].The choice for the associated function B s is not unique, and also not trivial to be ableto take into account the noise source term, which needs to be regularized. We present twofamilies of explicit stabilized integrators.For the first family, as proposed in [2], we use the second kind Chebyshev polynomialsdefined by T ′ s ( x ) = sU s − ( x ) and sin( sθ ) = sin( θ ) U s − (cos θ ) for all s ≥
0. Then theSK-ROCK method [2] is obtained with choosing A s , B s as follows: for s ≥
1, set A s ( z ) = T s ( ω + ω z ) T s ( ω ) , B s ( z ) = U s − ( ω + ω z ) U s − ( ω ) (cid:0) ω z (cid:1) , (19)where ω , ω are defined in(18). It is shown in [2] that the nearly optimal stability domainsize L s ≥ (2 − η ) s also persists for the size of the mean-square stability domain of SK-ROCK in the context of stiff SDEs, taking advantage of the classical identity on first andsecond kind Chebyshev polynomials, T s ( x ) + U s − ( x ) (1 − x ) = 1 . To illustrate the behavior of these polynomials, in Figure 1, we plot the polynomials A s ( z )and B s ( z ) in (19) as a function of real negative z , and we observe that for s = 7 , η = 0 . L s in (16) is close to the optimal value 2 s =98. In addition, the polynomial A s ( z ) oscillates between the values ± /T s ( ω ) where1 /T s ( ω ) ≃ − η = 0 .
95, while B s ( z ) oscillates with a very small amplitude. We alsoinclude for comparison the polynomial ( A s ( z ) − /z discussed below as an alternativedefinition of B s ( z ), and the stability function 1 / (1 − z ) of the implicit Euler method.Since A s and B s are polynomials (with degree depending on s ), the integrators canbe implemented explicitly, in particular no diagonalization or preconditioning techniquesof the operator Λ h are required in practice, in constrat to implicit methods. In practice,to avoid a dramatic accumulation of round-off errors, computing the operators A s ( τ Λ h )and B s ( τ Λ h ) should note be performed naively as high degree polynomials (see e.g. [1]).9lternatively, taking advantage that first and second kind Chebyshev polynomials havesatisfy the same recursion relation: for all s ≥ T s +1 ( x ) = 2 xT s ( x ) − T s − ( x ) , U s +1 ( x ) = 2 xU s ( x ) − U s − ( x ) , an efficient implementation inspired by [2] of the method (12) with stability polynomials A s , B s defined in (19) can then be achieved as follows. The first method considered in thepaper then writes: given u hn , compute u hn +1 by induction as K hn, = u hn , G hn = P h (cid:0) τ F ( u hn ) + σ ∆ W Qn (cid:1) ,K hn, = K hn, + τ µ Λ h (cid:0) K hn, + ν G hn (cid:1) + κ G hn ,K hn,i = µ i τ Λ h K hn,i − + ν i K hn,i − + κ i K hn,i − , i = 2 , . . . , s,u hn +1 = K hn,s , (20)with µ = ω /ω , ν = sω / κ = sµ , and for all i = 2 , . . . , s , µ i = 2 ω T i − ( ω ) T i ( ω ) , ν i = 2 ω T i − ( ω ) T i ( ω ) , κ i = − T i − ( ω ) T i ( ω ) = 1 − ν i . (21)We now describe the second family of explicit stabilized integrators considered in thisarticle: the polynomials A s and B s are given by A s ( z ) = T s ( ω + ω z ) T s ( ω ) , B s ( z ) = A s ( z ) − z , (22)a choice considered in [12] in the context of convex optimisation. Note that the twomethods use the same definition of A s . The second method considered in this paper canbe implemented as follows: given u hn , compute u hn +1 by induction as K hn, = u hn , G hn = P h (cid:0) τ F ( u hn ) + σ ∆ W Qn (cid:1) ,K hn, = K hn, + µ (cid:0) τ Λ h K hn, + G hn (cid:1) ,K hn,i = µ i (cid:0) τ Λ h K hn,i − + G hn (cid:1) + ν i K hn,i − + κ i K hn,i − , i = 2 , . . . , s,u hn +1 = K hn,s , (23)where µ = ω /ω and µ i , ν i , κ i , i = 2 , . . . , s are defined in (21). We are now in position to state the general assumptions and convergence results of thisarticle. The analysis of convergence of the integrators (12) is performed under the followingabstract conditions for the family of functions (cid:0) A s (cid:1) s ≥ and (cid:0) B s (cid:1) s ≥ . Assumption 3.3.
For all s ≥ , A s and B s are meromorphic functions, and for somesequence (cid:0) L s (cid:1) s ≥ of positive real numbers, the conditions below are satisfied. A s (0) = A ′ s (0) = B s (0) = 1 , for all s ≥ , (24) In addition sup s ≥ ,z ∈ [ − L s , (1 + | z | ) |B s ( z ) | < ∞ , (25)10 nd for all δ ∈ (0 , , sup s ≥ ,z ∈ [ − L s , − δ ] |A s ( z ) | < . (26) Finally, there exists δ ∈ (0 , such that sup s ≥ ,z ∈ D (0 ,δ ) (cid:0) |A s ( z ) | + |B s ( z ) | (cid:1) < ∞ , (27) where D (0 , δ ) = { z ∈ C ; | z | < δ } denotes the open disc of radius d and center in C . The strong order of convergence of the explicit stabilized methods (12) for the temporaldiscretization with respect to τ is equal to α/
2, analogously to the implicit Euler method(3). The strong order of convergence α/ Theorem 3.4. (Strong convergence in time)
Let Assumptions 2.4 and 3.3 be satisfied.For all α ∈ [0 , α ) and T ∈ (0 , ∞ ) , there exists C α,T ∈ (0 , ∞ ) such that for anyinitial condition u ∈ H and all h ∈ (0 , , τ ∈ (0 , and s ≥ such that the stabilitycondition (15) holds, then for all t n = nτ ≤ T one has (cid:0) E | u h ( t n ) − u hn | (cid:1) ≤ C α,T (1 + | u | t − α n ) τ α . (28)We deduce the strong convergence rates both in time and space, taking into accountthe finite element spatial discretization. This is an immediate consequence of Theorem 3.4,classical finite element discretization estimates (see Proposition A.2 in Appendix) and thetriangular inequality (cid:0) E | u ( t n ) − u hn | (cid:1) ≤ (cid:0) E | u h ( t n ) − u hn | (cid:1) + (cid:0) E | u ( t n ) − u h ( t n ) | (cid:1) . Corollary 3.5. (Strong convergence in time and space)
Let Assumptions 2.4 and 3.3be satisfied.For all α ∈ [0 , α ) and T ∈ (0 , ∞ ) , there exists C α,T ∈ (0 , ∞ ) such that for anyinitial condition u ∈ H and all h ∈ (0 , , τ ∈ (0 , and s ≥ such that the stabilitycondition (15) holds, then for all t n = nτ ≤ T one has (cid:0) E | u ( t n ) − u hn | (cid:1) ≤ C α,T (1 + | u | t − α n ) (cid:0) τ α + h α (cid:1) , (29) for all t n = nτ ≤ T . Remark 3.6.
The framework of our analysis includes not only explicit stabilized methodsbut also more general explicit or implicit Runge-Kutta type linearized methods. Indeed,observe that the conditions stated in Assumption 3.3 are satisfied if one considers theexplicit Euler method (for all s ≥ , set A s ( z ) = 1 + z , B s ( z ) = 1 ), with the constraint L s < , or the implicit Euler method (for all s ≥ , set A s ( z ) = B s ( z ) = (1 − z ) − ), with L s = ∞ . However, note that the Crank-Nicolson method (for all s ≥ , set A s ( z ) = z − z , B s ( z ) = − z ) satisfies (24) , (25) and (27) , with L s = ∞ , however the condition (26) isnot satisfied since |A s ( ∞ ) | = 1 (the method is A -stable but not L -stable, see [13]). F = 0 for simplicity (the case of a non-zero F is discussed in Remark4.7). Note that this corresponds formally to α = 2 in Assumption 2.4 and the statementof Theorem 3.7, where τ α/ can be replaced by τ − ǫ for arbitrarily small ǫ >
0. Note thatproving this result requires different techniques from the proof of Theorem 3.4.
Theorem 3.7. (Higher-order of convergence for spatially regular additive noise)
Under the assumptions of Theorem 3.4, assume in addition that sup s ≥ ,z ∈ [ − L s , min(1 , | z | )1 − A s ( z ) < ∞ . (30) Consider (1) with F = 0 and assume k ( − Λ) Q k L < ∞ . (31) For all ǫ ∈ (0 , and T ∈ (0 , ∞ ) , there exists C ǫ,T ∈ (0 , ∞ ) such that for any initial con-dition u ∈ H and all h ∈ (0 , , τ ∈ (0 , and s ≥ such that the stability condition (15) holds, then for all t n = nτ ≤ T one has (cid:0) E | u h ( t n ) − u hn | (cid:1) ≤ C ǫ,T (1 + | u | t − ǫn ) τ − ǫ . (32)Finally, in the deterministic case ( σ = 0), the explicit-stabilized scheme has orderof convergence in time equal to 1 and order of convergence in space equal to 2 in thedeterministic case σ = 0. The proof is omitted since it follows from straightforwardmodifications of the proof of Theorem 3.4, using the H¨older regularity of the solution withorder 1 − ǫ when σ = 0. Theorem 3.8. (Deterministic case)
Let Assumption 3.3 be satisfied.For all ǫ ∈ (0 , and T ∈ (0 , ∞ ) , there exists C ǫ,T ∈ (0 , ∞ ) such that for any ini-tial condition u ∈ H and all h ∈ (0 , , τ ∈ (0 , and s ≥ such that the stabilitycondition (15) holds, then for all t n = nτ ≤ T one has | u ( t n ) − u hn | ≤ C ǫ,T (1 + | u | t − ǫn ) (cid:0) τ − ǫ + h − ǫ (cid:1) . (33) This section is dedicated to the convergence analysis of the considered class of explicitstabilized methods for semilinear parabolic SPDEs. We first prove that the abstractconditions (Assumption 3.3) are indeed satisfied by the considered SK-ROCK methodand its variant both described in Section 3.3. We then derive the convergence analysis,based on general consistency results, moment estimates, spatial discretization estimates.12 .1 Verification of the abstract conditions for the Chebyshev methods
The goal of this section is to prove Proposition 4.2 below, which states that the two explicit-stabilized methods (20) and (23), with stability functions (19) and (22), respectively,satisfy the conditions given in Assumption 3.3.First, we recall some useful asymptotic results used to analyze properties of the explicit-stabilized integrators based on the first and second-kind Chebyshev polynomials, see [2].Recall that η is the damping parameter, see (18). Proposition 4.1.
Let η > . Then, when s → ∞ , one has ω s → s →∞ tanh( √ η ) √ η =: Ω( η ) , and for all δ ∈ [0 , η/ Ω( η )] , T s ( ω − ω δ ) → s →∞ cosh (cid:0)p η − δ Ω( η )) (cid:1) > . Let η = inf { η > η/ Ω( η ) = 1 } . Observe that η >
0, and that η/ Ω( η ) < η ∈ (0 , η ). The value of η has been estimated numerically: η ≃ . Proposition 4.2.
Let (cid:0) A s (cid:1) s =0 , ,... be defined by (17) and (cid:0) B s (cid:1) s =0 , ,... be defined eitherby (19) or by (22) , with the parameters given by (18) . Assume that η ∈ (0 , η ) . ThenAssumption 3.3 is satisfied, with L s = 2 ω − .Proof of Proposition 4.2. Note that A s and B s are polynomial, thus they are analytic functions on C . Proof of (24) . This follows from straightforward computations.
Proof of (27) . We recall the following formula for all s ≥ , k ≥ T ( k ) s (1) = k − Y j =0 s − j j + 1 ≤ s k . (34)Moreover, for all x ∈ [1 , + ∞ ), one has T ( k ) s ( x ) ≥
0, because all the roots of the polynomial T s (and of its derivatives T ( k ) s ) belong to the interval ( − , δ = 1. Let s ≥
1, then T s is a polynomial of degree s , and the Taylor formulayields, for all z ∈ C , A s ( z ) = T s ( ω + ω z ) T s ( ω ) = s X k =0 k ! T s ( ω ) T ( k ) s (1)( η/s + ω z ) k . Since the function T s is increasing on [1 , + ∞ ) and ω ≥
1, one has T s ( ω ) ≥ T s (1) = 1.In addition, T ′ s is increasing on [1 , ∞ ), thus using T ′ s ( ω ) ≥ T ′ s (1) = s , we obtain ω s ≤ ω T ′ s ( ω ) = T s ( ω ). Finally, using (34) then yieldssup s ≥ ω s ≤ sup s ≥ s X k =0 T ( k ) s (1) s k η k k ! ≤ e η . (35)13hese properties and the estimate (34) yieldsup z ∈ D (0 , |A s ( z ) | ≤ s X k =0 k ! (cid:0) η + ω s (cid:1) k ≤ e η + ω s ≤ e η + e η . To obtain an upper bound for sup z ∈ D (0 , |B s ( z ) | , the two versions need to be treated separately.First, assume that B s is defined by (19). Using the relation T ′ s = sU s − , the inequality T ′ s ( ω ) ≥ T ′ s (1) = s , and the upper bound | ω z | ≤ ω ≤ e η ≤ e η when z ∈ D (0 , s ≥ z ∈ D (0 , |B s ( z ) | ≤ e η s − X k =0 k ! T ′ s ( ω ) T ( k +1) s (1)( η/s + ω ) k ≤ e η s − X k =0 k ! ( η + ω s ) k ≤ e η + e η , where we used a Taylor expansion for the polynomial U s − ( ω + ω z ).Second, assume that B s is defined by (22). Observe that B s is an analytic function(since A s (0) = 1), thus using the maximum principle yields for all s ≥ z ∈ D (0 , |B s ( z ) | = sup | z | =1 (cid:12)(cid:12)(cid:12) A ( z ) − z (cid:12)(cid:12)(cid:12) ≤ z ∈ D (0 , |A s ( z ) | ≤ e η + e η . This concludes the proof of (27) for the choice δ = 1. Proof of (26) . For a fixed η , observe that it is sufficient to prove (26) for all δ ∈ (0 , ηe − η ].Indeed, the result then immediately follows as the left-hand side of (26) is by definitiona decreasing function of δ . We now assume δ ≤ ηe − η . Using (35) yields δ ≤ η/ ( ω s ) =( ω − /ω . On the one hand, if z ∈ [ − L s , (1 − ω ) /ω ], then x = ω + ω z ∈ [ − , | T s ( ω + ω z ) | ≤
1. On the other hand, if z ∈ [(1 − ω ) /ω , − δ ], then x = ω + ω z ≥ ≤ T s ( ω + ω z ) ≤ T s ( ω − ω δ ) < T s ( ω ), since T s is increasing on [1 , + ∞ ). Thus,sup z ∈ [ − L s , − δ ] |A s ( z ) | ≤ max(1 , T s ( ω − ω δ )) T s ( ω ) = A s ( − δ ) ∈ (0 , , for all s ≥ . Moreover, using Proposition 4.1, A s ( − δ ) → s →∞ cosh (cid:0)p η − δ Ω( η )) (cid:1) cosh (cid:0) √ η (cid:1) ∈ (0 , . One thus obtains the inequality sup s ≥ A s ( − δ ) ∈ (0 , Proof of (25) . Owing to (27), which holds with δ = 1, it is sufficient to provesup s ≥ ,z ∈ [ − L s , − | z ||B s ( z ) | < ∞ . It is again necessary to treat separately the two versions of the integrator.First, assume that B s is defined by (19). Recall that L s = 2 ω − , thus if z ∈ [ − ω − , − x = ω + ω z ∈ [ − η/s , ω − ω ]. Define Q s ( x ) = s ω T s ( ω ) (cid:18) x − η/s (cid:19) − ( x − η/s ) − x , − z B s ( z ) = U s − ( x ) (1 − x ) Q s ( x ) . Let us first prove the following claim: for all x ∈ [ − η/s , ω − ω ], one has Q s ( x ) ≥ s ≥ sup x ∈ [ − η/s ,ω − ω ] Q s ( x ) < ∞ . (36)Indeed, owing to (35), s ω T s ( ω ) ≤ e η . In addition, for all z ∈ [ − ω − , − x − η/s ω z ∈ [0 , . To treat the last term, note that for all x ∈ R , ddx (cid:18) − ( x − η/s ) − x (cid:19) = 2 η x − η/s xs (1 − x ) ≥ , where we used η/s ≤ η ≤
2. As a consequence, for − ηs ≤ x ≤ ω − ω = 1 + ηs − ω ,one obtains for all s ≥ ≤ Q s ( x ) ≤ e η − ( x − η/s ) − x ≤ sup s ≥ e η − (1 − ω ) − ( ω − ω ) < ∞ . Here, we used the property e η − (1 − ω ) − ( ω − ω ) → s →∞ e η − η/ Ω( η ) , and an asymptotic expansion with ω ∼ s →∞ Ω( η ) s − (owing to Proposition 4.1), ω = 1+ ηs ,and the property η/ Ω( η ) < η < η . This concludes the proof of (36).Since Q s ( x ) ≥ x ∈ [ − η/s , ω − ω ], and since z ≤
0, one obtains that U s − ( x ) (1 − x ) ≥
0. In addition, properties of the first and second-kind Chebyshevpolynomials yield the equality U s − ( x ) (1 − x ) = 1 − T s ( x ) ≤ x ∈ R . As aconsequence, sup s ≥ sup z ∈ [ − ω − , − − z B s ( z ) ≤ sup s ≥ sup x ∈ [ − η/s ,ω − ω ] Q s ( x ) < ∞ . This concludes the proof of (25) for the first version (19).Let us now focus on the second version: assume that B s is defined by (22). Then,using ( A s ( z ) − ≤ A s ( z ) ,sup s ≥ sup z ∈ [ − ω − , − − z B s ( z ) = sup s ≥ sup z ∈ [ − ω − , − |A s ( z ) − | | z | ≤ s ≥ sup z ∈ [ − ω − , − |A s ( z ) | < ∞ , owing to (26) with δ = 1.This concludes the proof of (25) for the two versions. The proof of Proposition 4.2 isthus completed. (cid:3) Proposition 4.3.
Let (cid:0) A s (cid:1) s =0 , ,... be defined by (17) with η ∈ (0 , η ) . Then (30) holds.Proof. Let δ ∈ (0 , ηe − η ). First, (26) yieldssup s ≥ ,z ∈ [ − L s , − δ ] min(1 , | z | )1 − A s ( z ) ≤ − sup s ≥ sup z ∈ [ − L s , − δ ] |A s ( z ) | < ∞ . Second, let z ∈ [ − δ,
0] and x = ω + ω z . Using δ ≤ ηe − η ≤ inf s ≥ ω − ω (owing to (35)),then one has x ≥ ω − ω δ ≥
1. Thus, A s ( z ) and A ′ s ( z ) are positive and increasing on theinterval [ − δ, A s ( z ) − z = Z A ′ s ( tz ) dt ≥ Z A ′ s ( − tδ ) dt = 1 − A s ( − δ ) δ , and noting 1 − A s ( z ) ≥ − A s ( z ), we obtainsup s ≥ ,z ∈ [ − δ, min(1 , | z | )1 − A s ( z ) ≤ δ − sup s ≥ A s ( − δ ) < ∞ , where we used (26). This concludes the proof of Proposition 4.3. (cid:3) The objective of this section is to state and prove some consequences of the abstractconditions given in Assumption 3.3. We start with the following lemma, which states that B s ( z ) = 1 + O ( z ) and A s ( z ) n = e nz + O ( n − ), uniformly for z ∈ [ − L s ,
0] and s ≥ Lemma 4.4.
Let Assumption 3.3 be satisfied. Then sup s ≥ ,z ∈ [ − L s , | − B s ( z ) || z | < ∞ , (37) and sup n ≥ ,s ≥ ,z ∈ [ − L s , n (cid:12)(cid:12) A s ( z ) n − e nz (cid:12)(cid:12) < ∞ . (38)The estimate (38) is inspired from [10, Theorem 9] for the time discretization ofparabolic PDEs and from [5, Lemma 5.2] in the context of explicit stabilized methodsfor parabolic homogenization problems. Proof of Lemma 4.4.
Let r s ( z ) = (1 − B s ( z )) /z . Since B s is a meromorphic function (byAssumption 3.3) with B s (0) = 1 (owing to (24)) and which is bounded on D (0 , δ ), then r s is an analytic function in the open disc D (0 , δ ), where δ is given by (27) (recall that δ does not depend on s ). Owing to the maximum principle for analytic functions,sup | z |≤ δ | r s ( z ) | = max θ ∈ [0 , π ] | r s ( δe iθ ) | ≤ Cδ
16n addition, using (25), we deducesup z ∈ [ − L s , − δ ] | r s ( z ) | ≤ δ − (cid:0) z ∈ [ − L s , − δ ] |B s ( z ) | (cid:1) < ∞ . This concludes the proof of (37).It remains to prove (38). For n ≥ s ≥
1, set ϕ n,s ( z ) = A s ( z ) n − e nz , and let r ,s ( z ) = ϕ ,s ( z ) /z . Since A s , and thus ϕ ,s , are meromorphic functions (by Assump-tion 3.3), with ϕ ,s (0) = ϕ ′ ,s (0) = 0 (owing to (24)), then r ,s is an analytic function inthe open disc D (0 , δ ), where δ is given by (27) (again, δ does not depend on s ). Owing tothe maximum principle for analytic functions, one has C = sup s ≥ , | z |≤ δ | r ,s ( z ) | = sup s ≥ sup | z | = δ | r ,s ( z ) | ≤ δ sup s ≥ , | z |≤ δ ( e δ + |A s ( z ) | ) < ∞ . Let γ ∈ (0 , δ ) be chosen sufficiently small such that γ + C γe γ/ < . Then, for all z ∈ [ − γ, |A s ( z ) | = e z/ | e z/ − e − z/ ϕ ,s ( z ) | ≤ e z/ (1 − | z | / | z | / C | z | e − z/ ) ≤ e z/ . Using the identity | ϕ n,s ( z ) | = |A s ( z ) n − e nz | = | z − ϕ ,s ( z ) || z n − X k =0 e kz A ( z ) n − k − | , one obtains for all n ≥ s ≥ ,z ∈ [ − γ, | ϕ n,s ( z ) | ≤ C | z | ne ( n − z/ ≤ n C e γ/ sup x ≥ (cid:0) x e − x/ (cid:1) . Let ρ = sup s ≥ ,z ∈ [ − L s , − γ ] |A s ( z ) | . Then ρ < n ≥
1, one hassup n ≥ ,s ≥ ,z ∈ [ − L s , − γ ] n | ϕ n,s ( z ) | ≤ sup n ≥ n (cid:0) e − nγ + ρ n (cid:1) < ∞ . Gathering the upper bounds for z ∈ [ − L s , − γ ] and z ∈ [ − γ,
0] concludes the proof of (38).The proof of Lemma 4.4 is thus completed. (cid:3)
Let us now apply the conditions given in Assumption 3.3 and obtained in Lemma 4.4,to deduce properties for the operators A s ( τ Λ h ) and B s ( τ Λ h ). Recall that k · k L ( V h ) denotesthe operator norm on L ( V h ). Proposition 4.5.
Let Assumption 3.3 be satisfied. For all γ ∈ [0 , , there exists C ∈ (0 , ∞ ) such that for all τ, h , and for s ≥ satisfying the stability condition (15) , then kA ( τ Λ h ) k L ( V h ) ≤ , (39) k ( − Λ h ) γ B ( τ Λ h ) k L ( V h ) ≤ Cτ γ , (40) k (cid:0) I − B ( τ Λ h ) (cid:1) ( − Λ h ) − γ k L ( V h ) ≤ Cτ γ , (41) kA ( τ Λ h ) n − e nτ Λ h k L ( V h ) ≤ Cn , for all n ≥ . (42)17 roof. The linear operators A ( τ Λ h ), B ( τ Λ h ), ( − Λ h ) γ , ( − Λ h ) − γ , and e nτ Λ h are self-adjointoperators, recall the notation (7). Owing to the stability condition (15), one has − τ λ m,h ∈ [ − L s ,
0] for all m ∈ { , . . . , N h } . Inequality (39) is then a straightforward consequenceof (26) (which holds for all arbitrarily small δ > γ = 0 and γ = 1), and of the following interpolation argument: usingthe inequalities B s ( z ) ≤ C and | z |B s ( z ) ≤ C , one obtains | z | γ B s ( z ) ≤ C − γ C γ , for γ ∈ [0 ,
1] and z = − τ λ m,h . Similarly, Inequality (41) is a consequence of (37) (in the case γ = 1). Finally, Inequality (42) follows from (38). (cid:3) We have obtained in the previous section all the ingredients for the proof of Theorem 3.4.The structure of the proof follows the same approach as for the convergence of the linearimplicit Euler scheme applied to parabolic semilinear SPDE discretization analysis withLipschitz continuous nonlinearities, see for instance [19]. Our proof however illustrates thebehavior of explicit-stabilized integrators in this context.In the proofs, the value of the constant C ∈ (0 , ∞ ) may change from line to line.The following estimate on the moments of the fully-discrete scheme is a key ingredientfor the proof of the convergence estimates. Proposition 4.6.
Let Assumptions 2.2, 2.4 and 3.1 be satisfied.Let (cid:0) u hn (cid:1) n ≥ be defined by (12) , where (cid:0) A s (cid:1) s ≥ and (cid:0) B s (cid:1) s ≥ satisfy Assumption 3.3.For all α ∈ [0 , α ) and T ∈ (0 , ∞ ) , there exists C α,T ∈ (0 , ∞ ) such that for all u ∈ H ,and all h ∈ (0 , , τ ∈ (0 , and s ≥ such that the stability condition (15) is satisfied,one has max nτ ≤ T E | ( − Λ h ) α u hn | ≤ C α,T (1 + | ( − Λ) α u | ) . In the proof of Theorem 3.4, only the result with α = 0 is used. Note that Propo-sition 4.6 shows that the spatial regularity is preserved by the temporal discretization,uniformly in the admissible parameters h, τ, s . Proof of Proposition 4.6.
By the Minkowski inequality, owing to the formulation (14) ofthe fully-discrete scheme, it is sufficient to deal with the three following contributions.(i) Owing to (39) and using Assumption 3.1, one has | ( − Λ h ) α A s ( τ Λ h ) n u h | ≤ | ( − Λ h ) α P h u | ≤ C | ( − Λ) α u | . (ii) Owing to the Minkowski inequality, and using the linear growth | F ( x ) | ≤ C (1 + | x | )(a consequence of the Lipschitz continuity of F ), one has (cid:16) E (cid:12)(cid:12) ( − Λ h ) α τ n − X k =0 A s ( τ Λ h ) n − − k B s ( τ Λ h ) P h F ( u hk ) (cid:12)(cid:12) (cid:17) ≤ τ n − X k =0 (cid:16) E (cid:12)(cid:12) ( − Λ h ) α A s ( τ Λ h ) n − − k B s ( τ Λ h ) P h F ( u hk ) (cid:12)(cid:12) (cid:17) ≤ Cτ n − X k =0 (cid:13)(cid:13) ( − Λ h ) α A s ( τ Λ h ) n − − k B s ( τ Λ h ) k L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) Cτ n − X k =0 (cid:13)(cid:13) ( − Λ h ) α e ( n − − k ) τ Λ h B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) + Cτ n − X k =0 (cid:13)(cid:13)(cid:0) A s ( τ Λ h ) n − − k − e ( n − − k ) τ Λ h (cid:1) ( − Λ h ) α B s ( τ Λ h ) k L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) . On the one hand, owing to (40) (applied with γ = α ), and to (10), one has τ n − X k =0 (cid:13)(cid:13) ( − Λ h ) α e ( n − − k ) τ Λ h B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) = τ (cid:13)(cid:13) ( − Λ h ) α B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) (cid:0) (cid:0) E | u hn − | (cid:1) (cid:1) + τ n − X k =0 (cid:13)(cid:13) ( − Λ h ) α e ( n − − k ) τ Λ h B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) ≤ Cτ − α (cid:0) (cid:0) E | u hn − | (cid:1) (cid:1) + Cτ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) α (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) . On the other hand, using first (42), then (40), τ n − X k =0 (cid:13)(cid:13)(cid:0) A s ( τ Λ h ) n − − k − e ( n − − k ) τ Λ h (cid:1) ( − Λ h ) α B s ( τ Λ h ) k L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) ≤ Cτ n − X k =0 n − − k ) (cid:13)(cid:13) ( − Λ h ) α B s ( τ Λ h ) k L ( V h ) (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) ≤ Cτ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) α (cid:0) (cid:0) E | u hk | (cid:1) (cid:1) . (iii) Using the Itˆo isometry formula, the condition (cid:13)(cid:13) ( − Λ h ) α + ǫ − P h Q k L ( H ) ≤ C , with ǫ ∈ (0 , α − α ) (see Proposition 3.2), and the inequalities (10), (40) and (42), combinedwith the same arguments as above, one obtains E (cid:12)(cid:12) ( − Λ h ) α n − X k =0 A s ( τ Λ h ) n − − k B s ( τ Λ h ) P h ∆ W Qk | = τ n − X k =0 (cid:13)(cid:13) ( − Λ h ) α A s ( τ Λ h ) n − − k B s ( τ Λ h ) P h Q (cid:13)(cid:13) L ( H ) ≤ τ n − X k =0 (cid:13)(cid:13) ( − Λ h ) − ǫ A s ( τ Λ h ) n − − k B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) (cid:13)(cid:13) ( − Λ h ) α + ǫ − P h Q k L ( H ) ≤ Cτ n − X k =0 (cid:13)(cid:13) ( − Λ h ) − ǫ e ( n − − k ) τ Λ h B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) Cτ n − X k =0 (cid:13)(cid:13) ( − Λ h ) − ǫ (cid:0) A s ( τ Λ h ) n − − k − e ( n − − k ) τ )Λ h (cid:1) B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) ≤ C (cid:0) τ ǫ + τ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) − ǫ (cid:1) + Cτ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) − ǫ (cid:13)(cid:13) ( − τ Λ h ) − ǫ B s ( τ Λ h ) (cid:13)(cid:13) L ( V h ) ≤ C. To conclude, first assume that α = 0. The estimate then follows from the applicationof the discrete Gronwall lemma. The case α ∈ (0 , α ) then follows from the calculationsabove. This concludes the proof of Proposition 4.6. (cid:3) Proof of Theorem 3.4.
Introduce the notation t n = nτ , n ∈ N , and assume that T = N τ ,with N ∈ N . Set F h = P h F .Let also ǫ n = (cid:0) E | u h ( nτ ) − u hn | (cid:1) . Using the mild formulations (9) and (14), oneobtains the decomposition u h ( t n ) − u hn = (cid:0) e nτ Λ h − A s ( τ Λ h ) n (cid:1) u h + n − X k =0 Z t k +1 t k (cid:2) e ( t n − t )Λ h − A s ( τ Λ h ) n − − k B s ( τ Λ h ) (cid:3) P h dW Q ( t )+ n − X k =0 Z t k +1 t k (cid:2) e ( t n − t )Λ h P h F ( u h ( t )) − A s ( τ Λ h ) n − − k B s ( τ Λ h ) F h ( u hk ) (cid:3) dt. Using Minkowski inequality, one obtains ǫ n ≤ ǫ n + ǫ n + ǫ n , where ǫ n = (cid:12)(cid:12)(cid:0) e nτ Λ h − A s ( τ Λ h ) n (cid:1) u h (cid:12)(cid:12) ǫ n = (cid:16)(cid:12)(cid:12)(cid:12) n − X k =0 E Z t k +1 t k h e ( t n − t )Λ h − A s ( τ Λ h ) n − − k B s ( τ Λ h ) i P h dW Q ( t ) (cid:12)(cid:12)(cid:12) (cid:17) ǫ n = n − X k =0 Z t k +1 t k (cid:0) E (cid:12)(cid:12) e ( t n − t )Λ h F h ( u h ( t )) − A s ( τ Λ h ) n − − k B s ( τ Λ h ) F h ( u hk ) (cid:12)(cid:12) (cid:1) dt. It remains to prove the three claims below: for all α ∈ [0 , α ), there exists a constant C such that ǫ n ≤ C | u | τ α t α n , (43) ǫ n ≤ Cτ α , (44) ǫ n ≤ Cτ n − X k =0 ǫ k + Cτ α (1 + | u | ) . (45) Proof of (43) . This claim follows from (42), indeed this inequality yields for all n ≥ | ǫ n | ≤ Cn | u h | ≤ C | u | τ α ( nτ ) α . roof of (44) . Using the Itˆo isometry formula,( ǫ n ) = n − X k =0 Z t k +1 t k (cid:13)(cid:13)h e ( t n − t )Λ h − A s ( τ Λ h ) n − − k B s ( τ Λ h ) i P h Q (cid:13)(cid:13) L ( H ) dt ≤ (cid:16) ( ǫ , n ) + ( ǫ , n ) + ( ǫ , n ) (cid:17) , where ( ǫ , n ) = Z t n (cid:13)(cid:13) e ( t n − t )Λ h ( I − B s ( τ Λ h )) P h Q (cid:13)(cid:13) L ( H ) dt ( ǫ , n ) = n − X k =0 Z t k +1 t k (cid:13)(cid:13)(cid:0) e ( t n − t )Λ h − e ( t n − t k +1 )Λ h (cid:1) B s ( τ Λ h ) P h Q (cid:13)(cid:13) L ( H ) dt ( ǫ , n ) = τ n − X k =0 (cid:13)(cid:13)(cid:0) e ( t n − t k +1 )Λ h − A s ( τ Λ h ) n − − k (cid:1) B s ( τ Λ h )) P h Q (cid:13)(cid:13) L ( H ) We next prove upper bounds for the quantities ( ǫ ,jn ) , j = 1 , , • Estimate of ǫ , n . Owing to Proposition 3.2, (10), and to (41),( ǫ , n ) ≤ Z t n (cid:13)(cid:13) e ( t n − t )Λ h ( I − B s ( τ Λ h ))( − Λ h ) − α − ǫ (cid:13)(cid:13) L ( V h ) dt (cid:13)(cid:13) ( − Λ h ) α + ǫ − P h Q (cid:13)(cid:13) L ( H ) ≤ C Z t n k ( − Λ h ) − ǫ e ( t n − t )Λ h k L ( V h ) dt k ( I − B ( τ Λ h ))( − Λ h ) − α k L ( V h ) ≤ C Z t n t ǫ − dtτ α ≤ Cτ α . • Estimate of ǫ , n . Owing to Proposition 3.2, (11), and to (40),( ǫ , n ) ≤ C n − X k =0 Z t k +1 t k k (cid:0) e ( t k +1 − t )Λ h − I )( − Λ h ) − α k L ( V h ) dt k e ( t n − t k +1 )Λ h ( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) + C Z t n t n − k (cid:0) e ( t n − t )Λ h − I )( − Λ h ) − α (cid:13)(cid:13) L ( V h ) dt (cid:13)(cid:13) B s ( τ Λ h )( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) ≤ Cτ α (cid:0) τ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) − ǫ + τ ǫ (cid:1) ≤ Cτ α . • Estimate of ǫ , n . Owing to Assumption 2.4, (42), and to (40),( ǫ , n ) ≤ Cτ n − X k =0 k (cid:0) e ( t n − t k +1 )Λ h − A s ( τ Λ h ) n − − k (cid:1) k L ( V h ) kB s ( τ Λ h )( − Λ h ) − ǫ − α k L ( V h ) ≤ Cτ n − X k =0 n − − k ) − ǫ τ α + ǫ − Cτ α (cid:0) τ n − X k =0 (cid:0) ( n − − k ) τ (cid:1) − ǫ (cid:1) ≤ Cτ α . Proof of (45) . The error term ǫ n is decomposed as follows: ǫ n ≤ ǫ , n + ǫ , n + ǫ , n + ǫ , n , where ǫ , n = n − X k =0 Z t k +1 t k (cid:0) E (cid:12)(cid:12) e ( t n − t )Λ h (cid:0) F h ( u h ( t )) − F h ( u hk ) (cid:1)(cid:12)(cid:12) (cid:1) dtǫ , n = n − X k =0 Z t k +1 t k (cid:0) E (cid:12)(cid:12) e ( t n − t )Λ h (cid:0) I − B s ( τ Λ h ) (cid:1) F h ( u hk ) (cid:12)(cid:12) (cid:1) dtǫ , n = n − X k =0 Z t k +1 t k (cid:0) E (cid:12)(cid:12)(cid:0) e ( t n − t )Λ h − e ( t n − t k +1 )Λ h (cid:1) B s ( τ Λ h ) F h ( u hk ) (cid:12)(cid:12) (cid:1) dtǫ , n = τ n − X k =0 (cid:0) E (cid:12)(cid:12)(cid:0) e ( t n − t k +1 )Λ h − A s ( τ Λ h ) n − − k (cid:1) B s ( τ Λ h ) F h ( u hk ) (cid:12)(cid:12) (cid:1) . We next estimate the quantities ǫ , n , j = 1 , , , • Estimate of ǫ , n . By the Lipschitz continuity of F h (uniformly with respect to theparameter h ∈ (0 , ǫ , n ≤ Cτ n − X k =0 (cid:0) E | u h ( t k ) − u hk | (cid:1) + C n − X k =0 Z t k +1 t k (cid:0) E | u h ( t ) − u h ( t k ) | (cid:1) dt ≤ Cτ n − X k =0 ǫ k + Cτ α (cid:0) τ − α + τ n − X k =1 (1 + | u h | ( kτ ) α ) (cid:1) . • Estimate of ǫ , n . Using the linear growth property of F , Proposition 4.6, and theinequalities (41) and (10), one has ǫ , n ≤ C k ( − Λ h ) − α ( I − B s ( τ Λ h )) k L ( V h ) n − X k =0 Z t k +1 t k k e ( t n − t )Λ h ( − Λ h ) α k L ( V h ) dt (cid:0) E | u hk | (cid:1) ≤ Cτ α Z t n dt ( t n − t ) α (1 + | u | ) . • Estimate of ǫ , n . Using the linear growth property of F , Proposition 4.6, inequal-ity (40), and using a combination of (10) and (11), then one obtains ǫ , n ≤ C (1 + | u | ) n − X k =0 Z t k +1 t k (cid:13)(cid:13) e ( t n − t )Λ h − e ( t n − t k +1 )Λ h (cid:13)(cid:13) L ( V h ) dt C (1 + | u | ) n − X k =0 Z t k +1 t k (cid:13)(cid:13)(cid:0) e ( t k +1 − t )Λ h − I (cid:1) e ( t n − t k +1 )Λ h (cid:13)(cid:13) L ( V h ) dt ≤ C (1 + | u | ) τ α (cid:0) τ − α + τ n − X k =0 C (cid:0) ( n − − k ) τ (cid:1) α (cid:1) . • Estimate of ǫ , n . Using the linear growth property of F , and the inequality (40), andthen using (42), one obtains ǫ , n ≤ C (1 + | u | ) τ n − X k =0 (cid:13)(cid:13) e ( t n − t k +1 )Λ h − A s ( τ Λ h ) n − − k (cid:13)(cid:13) L ( V h ) ≤ C (1 + | u | ) τ α (cid:0) τ − α + τ n − X k =0 C (cid:0) ( n − − k ) τ (cid:1) α (cid:1) . Combining the error estimates (43), (44) and (45), the error satisfies ǫ = 0 and ǫ n ≤ Cτ n − X k =0 ǫ k + Cτ α (1 + t − α n | u | ) . Applying the discrete Gronwall lemma yields the result and this concludes the proof ofTheorem 3.4. (cid:3)
The proof of Theorem 3.7 requires a different approach from the proof above of Theo-rem 3.4 to obtain order 1 of strong convergence for the time discretization of the SPDE (1)driven by additive noise (instead of order at most 1 / Proof of Theorem 3.7.
Let us first establish that there exists C ∈ (0 , ∞ ) such that for all s ≥ z ∈ [ − L s ,
0] and n ≥
1, one has |A s ( z ) n − e nz | ≤ C min(1 , n | z | )( |A ( z ) | n − + e ( n − z ) . (46)On the one hand, sup s ≥ ,z ∈ [ − L s , |A ( z ) | < ∞ by (26), thus |A s ( z ) n − e nz | ≤ |A s ( z ) | n + e nz ≤ C |A s ( z ) | n − + e ( n − z . On the other hand, using b m − a m ≤ mb m − ( b − a ) for all 0 ≤ a ≤ b and all m ≥
1, wededuce |A s ( z ) n − e nz | ≤ n |A s ( z ) − e z | ( |A s ( z ) | n − + e ( n − z ) ≤ n | z | | r ,s ( z ) | ( |A s ( z ) | n − + e ( n − z ) , where r ,s ( z ) = A s ( z ) − e z z . We use the same techniques as in the proof of Lemma 4.4. Let δ ∈ (0 , s ≥ ,z ∈ [ − δ, | r ,s ( z ) | < ∞ .
23n addition, using (26), one hassup s ≥ ,z ∈ [ − L s , − δ ] | r ,s ( z ) | ≤ δ ( sup s ≥ ,z ∈ [ − L s , − δ ] |A s ( z ) | + e − δ ) < ∞ . This concludes the proof of (46).It remains to prove the error estimate (32). Note that, due to Assumption 3.1, onehas the following variant of the result of Proposition 3.2,sup h ∈ (0 , k ( − Λ h ) P h Q (cid:13)(cid:13) L ( H ) ≤ k ( − Λ) Q k L ( H ) < ∞ , where we have used the assumption (31).We then have the following decomposition of the error, which is analogous to that inthe proof of Theorem 3.4, but with ǫ n = 0 due to F = 0, (cid:0) E | u h ( nτ ) − u hn | (cid:1) ≤ ǫ n + ǫ n , with ǫ n ≤ (cid:0) ( ǫ , n ) + ( ǫ , n ) + ( ǫ , n ) (cid:1) . The term ǫ n is treated using (42): | ǫ n | ≤ Cn | u h | ≤ Cτ | u | t n . In addition, using the inequalities from Proposition 4.5, the terms ǫ , n and ǫ , n can betreated similarly:( ǫ , n ) = Z t n (cid:13)(cid:13) e ( t n − t )Λ h ( I − B s ( τ Λ h )) P h Q (cid:13)(cid:13) L ( H ) dt ≤ Z t n (cid:13)(cid:13) e ( t n − t )Λ h ( I − B s ( τ Λ h ))( − Λ h ) − (cid:13)(cid:13) L ( V h ) dt (cid:13)(cid:13) ( − Λ h ) P h Q (cid:13)(cid:13) L ( H ) ≤ C Z t n (cid:13)(cid:13) ( − Λ h ) − ǫ e t Λ h (cid:13)(cid:13) L ( V h ) dt (cid:13)(cid:13) ( I − B s ( τ Λ h ))( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) ≤ C ( T ) τ − ǫ ) , and analogously,( ǫ , n ) = n − X k =0 Z t k +1 t k (cid:13)(cid:13)(cid:0) e ( t n − t )Λ h − e ( t n − t k +1 )Λ h (cid:1) B s ( τ Λ h ) P h Q (cid:13)(cid:13) L ( H ) dt ≤ C n − X k =0 Z t k +1 t k k (cid:0) e ( t k +1 − t )Λ h − I )( − Λ h ) − ǫ k L ( V h ) dt k e ( t n − t k +1 )Λ h ( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) + C Z t n t n − k (cid:0) e ( t n − t )Λ h − I )( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) dt (cid:13)(cid:13) B s ( τ Λ h )( − Λ h ) − ǫ (cid:13)(cid:13) L ( V h ) ≤ C (cid:0) τ n − X k =0 τ − ǫ ) (cid:0) ( n − − k ) τ (cid:1) − ǫ + τ (cid:1) .
24t remains to deal with ǫ , n , using different arguments from the proof of Theorem 3.4.( ǫ , n ) = τ n − X k =0 (cid:13)(cid:13)(cid:0) e ( t n − t k +1 )Λ h − A s ( τ Λ h ) n − − k (cid:1) B s ( τ Λ h )) P h Q (cid:13)(cid:13) L ( H ) ≤ τ ∞ X k =0 N h X m =1 | Q e m,h | (cid:12)(cid:12) e − kτλ m,h − A s ( − τ λ m,h ) k (cid:12)(cid:12) |B s ( − τ λ m,h ) | ≤ Cτ N h X m =1 | Q e m,h | min(1 , τ λ m,h ) ∞ X k =0 k (cid:0) e − kτλ m,h + |A s ( − τ λ m,h ) | k (cid:1) , where we have used the claim (46).Since sup x ∈ [0 , (1 − x ) P ∞ k =0 k x k < ∞ and min(1 , y ) ≤ y min(1 , y ) for all y ≥
0, oneobtains( ǫ , n ) ≤ Cτ N h X m =1 | Q e m,h | min(1 , τ λ m,h ) (cid:0) − e − τλ m,h ) + 1(1 − |A s ( − τ λ m,h ) | ) (cid:1) ≤ Cτ N h X m =1 | Q ( − Λ h ) e m,h | (cid:0) min(1 , τ λ m,h ) (1 − e − τλ m,h ) + min(1 , τ λ m,h ) (1 − |A s ( − τ λ m,h | ) (cid:1) . In addition, from assumption (30), we deducesup s ≥ ,z ∈ [ − L s , (cid:0) min(1 , | z | )1 − e − z + min(1 , | z | )1 − |A s ( z ) | (cid:1) < ∞ . Finally, this yields( ǫ , n ) ≤ Cτ N h X m =1 | Q ( − Λ h ) e m,h | ≤ Cτ k ( − Λ) Q k L ( H ) . Gathering the above estimates concludes the proof of Theorem 3.7. (cid:3)
Remark 4.7.
The statement of Theorem 3.7 could be extended for a non zero F , assumingthat F is of class C from H to itself with bounded first and second derivatives. In theproof of the main Theorem 3.4 one would need to change the way ǫ , n is treated (withthis approach one can not overcome the order / ). Performing a second-order Taylorexpansion and using the stochastic Fubini theorem would give the result. The details arestandard and are omitted for brevity. In this section, we illustrate our convergence analysis on the following stochastic heatequation with additive space-time white noise in dimension d = 1, ∂ t u ( x, t ) = ∂ xx u ( x, t ) + f ( u ( x, t )) + g ( u ( x, t )) ∂ t W ( x, t ) , (47)25 -4 -3 -2 -1 stepsize -2 -1 s t r ong e rr o r SK-ROCKVariantImplicit Eulerslope 1/4 (a) Additive noise ((47), g ( u ) = 1). -4 -3 -2 -1 stepsize -3 -2 -1 s t r ong e rr o r SK-ROCKVariantImplicit Eulerslope 1/4 (b) Multiplicative noise ((48), g ( u ) = u ). Figure 2: Strong convergence plots using SK-ROCK (20), the variant (23), and the implicitEuler method (3) for the stochastic heat equation with space-time white noise, with finaltime T = 0 .
1. Averages over 10 samples.on the domain D = (0 ,
1) where we consider homogeneous Dirichlet boundary conditions u (0 , t ) = u (1 , t ) = 0 and the initial condition u ( x,
0) = sin(2 πx ). We also consider thestochastic heat equation with multiplicative space-time white noise in dimension d = 1 ∂ t u ( x, t ) = ∂ xx u ( x, t ) + f ( u ( x, t )) + u ( x, t ) ∂ t W ( x, t ) , (48)We use a standard finite difference method with constant mesh size h = 1 /
100 fordiscretizing the Laplacian and the white noise, and obtain the following system where u ( x i , t ) ≃ u i ( t ) with x i = ih, i = 1 , . . . N , h = 1 /N , du i = u i +1 − u i + u i − h dt + f ( u i ) dt + 1 √ h g ( u i ) dw i , i = 1 , . . . , N − , where w i , i = 1 , . . . N are independent Wiener processes, and u = u N = 0 to take intoaccount the homogeneous Dirichlet boundary conditions, with g ( u ) = 1 for (47) and g ( u ) = u for (48).In Figure 2, we plot the convergence curves for the strong error (cid:0) E | u h ( nτ ) − u hn | (cid:1) both for the additive noise case (47) (see Fig. 2(a)) and the multiplicative noise case (48)(see Fig. 2(b)) using the nonlinearity f ( u ) = − u − sin( u ). We used averages over 10 trajectories and a reference solution was computed with the small timestep τ = 0 . · − .We observe in Figure 2 lines of slope 1/4, this corroborates the strong convergence estimateof Theorem 3.4 in the additive noise case and suggests that the strong convergence estimatepersists in the multiplicative noise case (recall that α = 1 / Note that such a standard finite difference method on a uniform mesh can be seen as the simplestfinite element method which is thus covered by our analysis. a) SK-ROCK method (20). (b) Variant method (23).(c) Linear implicit Euler method (3). x -0.200.20.40.60.81 u ( x , ) SK-ROCKVariantLinearized Euler (d) Solution profile at final time T = 1. Figure 3: Samples of realisation of the stochastic nonlinear heat equation (47) with non-linearity f ( u ) = − u − sin( u ) and additive noise g ( u ) = 1 using explicit stabilized methods(20), (23). With time and spatial stepsizes τ = h = 1 / f ( u ) = − u − sin( u ) using the two explicit stabilized methods (20), (23) and thelinear implicit Euler method (3), respectively. For comparison, we used the same randomrealizations for sampling the noise. We also plot in Figure 3(d)the corresponding profilesat final time T = 1. It can be observed that compared to the SK-ROCK method (20)the variant method (23) and the linear implicit Euler method (3) exhibit an increasedregularity. Note that increasing the damping parameter η of the explicit stabilized methodswould increase the regularity of the numerical solutions (this is not illustrated here forbrevity). We mention that the question of the spatial regularity of the numerical solutionis addressed in [9, Prop. 3.9], where it is shown that the same regularity as the exactsolution can be recovered by introducing a suitable postprocessor for the linear implicitEuler method applied to the stochastic heat equation. Acknowledgements.
The work of AA was partially supported by the Swiss NationalScience Foundation, project No. 200020 172710. The work of C.-E. B. was partiallysupported by the project SIMALIN (ANR-19-CE40-0016) operated by the French National27esearch Agency. The work of GV was partially supported by the Swiss National ScienceFoundation, projects No. 200020 184614 and No. 200020 178752. The computations wereperformed at the University of Geneva on the Baobab cluster.
A Appendix
In this appendix, we first provide spatial and temporal regularity properties on the process (cid:0) u ( t ) (cid:1) t ≥ and its semi-discrete approximation (cid:0) u h ( t ) (cid:1) t ≥ . The proofs are omitted sincethey are quite standard and can be found for instance in [11] or [16]. Let us first recall thefollowing well-posedness and time regularity results, see for instance [17, Chap. 2, Thm.2.31] and [18, Chap. 10, Thm. 10.26, Thm. 10.27], and [11]. Proposition A.1.
Let Assumptions 2.2 and 2.4 be satisfied.For any initial condition u ∈ H , there exists a unique global mild solution (cid:0) u ( t ) (cid:1) t ≥ of (1) : the process u is continuous with values in H and satisfies (6) for all t ≥ .Moreover, assume that | ( − Λ) α u | < ∞ , with α ∈ [0 , α ) (recall that α is defined inAssumption 2.4). For every α ∈ [ α , α ) and T ∈ (0 , ∞ ) , there exists a constant C ∈ (0 , ∞ ) such that for all t ∈ (0 , T ] , E | ( − Λ) α u ( t ) | ≤ C (cid:0) | ( − Λ) α u | t α − α (cid:1) , and for all < t ≤ t ≤ T , E | u ( t ) − u ( t ) | ≤ C | t − t | α (cid:0) | ( − Λ) α u | t α − α (cid:1) . The well-posedness part of the result is obtained by a standard fixed point argumentand the computation below with α = 0. For the spatial regularity estimate, note that thestochastic contribution is treated as follows: by the Itˆo isometry formula, E | ( − Λ) α Z t e ( t − s )Λ dW Q ( s ) | = Z t (cid:13)(cid:13) ( − Λ) α e ( t − s )Λ Q (cid:13)(cid:13) L ( H ) ds ≤ Z t (cid:13)(cid:13) ( − Λ) − ǫ e ( t − s )Λ (cid:13)(cid:13) L ( H ) ds k ( − Λ) α + ǫ − Q k L ( H ) ≤ C Z t ( t − s ) ǫ − ds k ( − Λ) α + ǫ − Q k L ( H ) , where ǫ ∈ (0 , α − α ). In addition, the smoothing properties (5) of the semi-group yields (cid:12)(cid:12) ( − Λ) α e t Λ u (cid:12)(cid:12) ≤ k ( − Λ) α − α e t Λ k L ( H ) | ( − Λ) α u | ≤ Ct − α − α | ( − Λ) α u | . For the temporal regularity estimate, similar arguments are used to prove that, first, (cid:12)(cid:12) Z t e ( t − s )Λ dW Q ( s ) − Z t e ( t − s )Λ dW Q ( s ) (cid:12)(cid:12) ≤ C | t − t | α , and, second, using the smoothing properties (5), | (cid:0) e t Λ − e t Λ (cid:1) u | ≤ k (cid:0) e ( t − t )Λ − I (cid:1) e t Λ ( − Λ) − α k L ( H ) | ( − Λ) α u | ≤ C | t − t | α t α − α .
28e now state without proof the following estimate concerning the regularity propertiesof u h defined by (8), and the spatial discretization error. We refer for instance to [17,Chap. 2, Thm. 2.27], [18, Chap. 10, Thm. 10.28], and [16]. Proposition A.2.
Let Assumptions 2.2, 2.4 and 3.1 be satisfied. Assume | ( − Λ) α u | < ∞ , with α ∈ [0 , α ) . Then, for every α ∈ [ α , α ) and T ∈ (0 , ∞ ) , there exists C ∈ (0 , ∞ ) such that, for all t ∈ (0 , T ] , sup h ∈ (0 , E | ( − Λ) α h u h ( t ) | ≤ C (cid:0) | ( − Λ) α u | t α − α (cid:1) , and for all < t ≤ t ≤ T , sup h ∈ (0 , E | u h ( t ) − u h ( t ) | ≤ C | t − t | α (cid:0) | ( − Λ) α u | t α − α (cid:1) . For all α ∈ [ α , α ) and T ∈ (0 , ∞ ) , there exists C ∈ (0 , ∞ ) , such that for all t ∈ (0 , T ] , (cid:0) E (cid:12)(cid:12) u h ( t ) − u ( t ) (cid:12)(cid:12) (cid:1) ≤ Ch α (cid:0) | ( − Λ) α u | t α − α (cid:1) . Proposition A.2 states that the strong order of convergence in space is α , which istwice the order α/ u stated in Proposition A.1. References [1] A. Abdulle.
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