Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG k method
SSuperconvergent interpolatory HDG methods for reaction diffusionequations I: An HDG k method Gang Chen ∗ Bernardo Cockburn † John R. Singler ‡ Yangwen Zhang § Abstract
In our earlier work [8], we approximated solutions of a general class of scalar parabolicsemilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG)method. This method reduces the computational cost compared to standard HDG since theHDG matrices are assembled once before the time integration. Interpolatory HDG also achievesoptimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reactiondiffusion problems, and use the postprocessed approximate solution to evaluate the nonlinearterm. We prove this simple change restores the superconvergence and keeps the computationaladvantages of the Interpolatory HDG method. We present numerical results to illustrate theconvergence theory and the performance of the method.
Keywords
Interpolatory hybridizable discontinuous Galerkin method, superconvergence
In our earlier work [8], we introduced an interpolatory hybridizable discontinuous Galerkin (In-terpolatory HDG) method to approximate the solution of semilinear parabolic PDEs. In contrastto standard HDG, the Interpolatory HDG method uses an elementwise interpolation procedure toapproximate the nonlinear term; therefore, all quadrature for the nonlinear term can be performedonce before the time integration, which yields a significant computational cost reduction. TheInterpolatory HDG method still converged at optimal rates, but superconvergence using element-by-element postprocessing was lost.The superconvergence is an excellent feature of HDG methods, and therefore in this work wemodify the Interpolatory HDG method from [8] and restore the superconvergence for reactiondiffusion PDEs.Specifically, we consider the following class of scalar reaction diffusion PDEs on a Lipschitzpolyhedral domain Ω ⊂ R d , d = 2 ,
3, with boundary ∂ Ω: ∂ t u − ∆ u + F ( u ) = f in Ω × (0 , T ] ,u = 0 on ∂ Ω × (0 , T ] ,u ( · ,
0) = u in Ω . (1.1) ∗ School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu, China,[email protected] † School of Mathematics, University of Minnesota, Minneapolis, MN, [email protected] ‡ Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, [email protected] § Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO,[email protected] a r X i v : . [ m a t h . NA ] M a y . Chen, B. Cockburn, J. Singler, and Y. ZhangIn Section 2, we provide background on HDG methods and describe the new Interpolatory HDGapproach in detail. We use the HDG k method to approximate the linear terms in the equation;i.e., k th order discontinuous polynomials are used to approximate the flux q = −∇ u , the scalarvariable u , and its trace, and the stabilization function is chosen as O (1) piecewise constant. Forthe nonlinear term, we again use an elementwise Lagrange interpolation operator, as in [8], but nowwe also approximate u using a postprocessing approach. This modified approximate nonlinearityrestores the superconvergence and, as in [8], we have a simple explicit expressions for the nonlinearterm and Jacobian matrix, which leads to an efficient and unified implementation.We analyze the semidiscrete Interpolatory HDG k method in Section 3. We first assume the non-linearity satisfies a global Lipschitz condition and prove the superconvergence. Next, we establishthe superconvergence under a local Lipschitz condition, assuming the mesh is quasi-uniform.In Section 4, we illustrate the convergence theory with numerical experiments and also demon-strate the performance of the Interpolatory HDG k method on a reaction diffusion PDE system.We note that interpolatory finite element methods for nonlinear PDEs are well-known to havecomputational advantages and have a long history. The approach has been given many differentnames, including finite element methods with interpolated coefficients, product approximation, andthe group finite element method. For more information, see [4, 6, 7, 17–20, 22–24, 32, 34–36, 38–41]and the references therein. k formulation and implementation Hybridizable discontinuous Galerkin (HDG) methods were proposed by Cockburn et al. in [13].HDG methods work with the mixed formulation of the PDE, and on each element the approximatesolution and flux are expressed in terms of the approximate solution trace on the element boundary.The approximate trace is uniquely determined by requiring the normal component of the numericaltrace of the flux to be continuous across element boundaries. This allows the approximate solutionand approximate flux variables to be eliminated locally on each element; the result is a globalsystem of equations for the approximate solution trace only. Therefore, the number of globallycoupled degrees of freedom for HDG methods is significantly lower than for standard DG methods.HDG methods have been successfully applied to linear PDEs [13–15, 27] and nonlinear PDEs [2, 16,21, 25, 26, 28–31].To describe the Interpolatory HDG k method, we introduce notation below. We mostly followthe notation used in [13], where HDG methods were considered for linear, steady-state diffusion.Let T h be a collection of disjoint simplexes K that partition Ω. Let ∂ T h denote the set { ∂K : K ∈ T h } . For an element K in the collection T h , let e = ∂K ∩ Γ denote the boundary face of K ifthe d − e is nonzero. For two elements K + and K − of the collection T h , let e = ∂K + ∩ ∂K − denote the interior face between K + and K − if the d − e is nonzero. Let ε oh and ε ∂h denote the sets of interior and boundary faces, respectively, and let ε h denote the union of ε oh and ε ∂h . We use the mesh-dependent inner products( w, v ) T h := (cid:88) K ∈T h ( w, v ) K , (cid:104) ζ, ρ (cid:105) ∂ T h := (cid:88) K ∈T h (cid:104) ζ, ρ (cid:105) ∂K , where ( · , · ) D denotes the L ( D ) inner product for a set D ⊂ R d and (cid:104)· , ·(cid:105) Γ denotes the L (Γ) innerproduct for a set Γ ⊂ R d − .Let P k ( D ) denote the set of polynomials of degree at most k on a domain D . We consider the2uperconvergent interpolatory HDG methods for reaction diffusion Idiscontinuous finite element spaces V h := { v ∈ [ L (Ω)] d : v | K ∈ [ P k ( K )] d , ∀ K ∈ T h } , (2.1) W h := { w ∈ L (Ω) : w | K ∈ P k ( K ) , ∀ K ∈ T h } , (2.2) Z h := { z ∈ L (Ω) : z | K ∈ P k +1 ( K ) , ∀ K ∈ T h } , (2.3) M h := { µ ∈ L ( ε h ) : µ | e ∈ P k ( e ) , ∀ e ∈ ε h , µ | ε ∂h = 0 } . (2.4)All spatial derivatives of functions in these spaces should be understood piecewise on each element K ∈ T h .We consider the HDG method that approximates the scalar variable u , flux q = −∇ u , andboundary trace (cid:98) u using the spaces W h , V h , and M h , respectively; i.e., polynomials of degree k areused for all variables. We call this specific method HDG k to distinguish it from the wide variety ofother available HDG methods, see, e.g., [9–12]. The space Z h is used for postprocessing.For the Interpolatory HDG k method, we use an elementwise interpolatory procedure alongwith postprocessing to approximate the nonlinear term. Let I h be the elementwise interpolationoperator with respect to the finite element nodes for the postprocessing space Z h . Therefore, forany function g that is continuous on each element we have I h g ∈ Z h .The Interpolatory HDG k formulation reads: find ( q h , u h , (cid:98) u h ) ∈ V h × W h × M h such that, for all( r h , v h , (cid:98) v h ) ∈ V h × W h × M h , we have( q h , r h ) T h − ( u h , ∇ · r h ) T h + (cid:104) (cid:98) u h , r h · n (cid:105) ∂ T h = 0 , (2.5a)( ∂ t u h , v h ) T h − ( q h , ∇ v h ) T h + (cid:104) (cid:98) q h · n , v h (cid:105) ∂ T h + ( I h F ( u (cid:63)h ) , v h ) T h = ( f, v h ) T h , (2.5b) (cid:104) (cid:98) q h · n , (cid:98) v h (cid:105) ∂ T h \ ε ∂h = 0 , (2.5c) u h (0) = Π u , (2.5d)where Π is a projection mapping into W h and the numerical trace for the flux is defined by (cid:98) q h · n = q h · n + τ ( u h − (cid:98) u h ) . (2.6)Here, the stabilization function τ is nonnegative, constant on each element, and O (1). Furthermore,the postprocessed scalar variable u (cid:63)h = q k +1 h ( q h , u h ) ∈ Z h is determined on each element K by( ∇ q k +1 h ( q h , u h ) , ∇ z h ) K = − ( q h , ∇ z h ) K , (2.7a)( q k +1 h ( q h , u h ) , w h ) K = ( u h , w h ) K , (2.7b)for all ( z h , w h ) ∈ [ P k +1 ( K )] ⊥ × P ( K ), where[ P k +1 ( K )] ⊥ = { z ∈ P k +1 ( K ) : ( z, w ) K = 0 for all w ∈ P ( K ) } . (2.8) Remark . In our original Interpolatory HDG work [8], we used I kh F ( u h ) to approximate thenonlinear term, where I kh is the elementwise interpolation operator mapping into W h . We provedoptimal convergence rates for all variables, but we did not observe superconvergence after anelement-by-element postprocessing. In this work, we approximate the nonlinearity using I h andpostprocessing, i.e., I h F ( u (cid:63)h ). Note that this approximate nonlinearity is in Z h instead of W h asin our first work. This simple change yields the superconvergence and keeps all the advantages ofthe original Interpolatory HDG k method proposed in [8]. We provide details on the computationaladvantages of this approach in Section 2.1. 3. Chen, B. Cockburn, J. Singler, and Y. Zhang In our original work [8] on Interpolatory HDG, we provided details of the implementation for themethod. Since we changed the discretization of the nonlinear term in this work, the implementationis different; therefore, we provide details for the implementation of this new formulation and showhow all matrices need only be assembled once before the time integration. As in our earlier work [8],we describe the implementation using a simple time discretization approach: backward Euler witha Newton iteration to solve the nonlinear system at each time step. Using Interpolatory HDG withother time discretization approaches is also possible.Let N be a positive integer and define the time step ∆ t = T /N . We denote the approximationof ( q h ( t ) , u h ( t ) , (cid:98) u h ( t )) by ( q nh , u nh , (cid:98) u nh ) at the discrete time t n = n ∆ t , for n = 0 , , , . . . , N . Wereplace the time derivative ∂ t u h in (2.5) by the backward Euler difference quotient ∂ + t u nh = u nh − u n − h ∆ t . (2.9)This gives the following fully discrete method: find ( q nh , u nh , (cid:98) u nh ) ∈ V h × W h × M h satisfying( q nh , r ) T h − ( u nh , ∇ · r ) T h + (cid:104) (cid:98) u nh , r · n (cid:105) ∂ T h = 0 , (2.10a)( ∂ + t u nh , w ) T h − ( q nh , ∇ w ) T h + (cid:104) (cid:98) q nh · n , w (cid:105) ∂ T h + ( I h F ( u n(cid:63)h ) , w ) T h = ( f n , w ) T h , (2.10b) (cid:104) (cid:98) q nh · n , µ (cid:105) ∂ T h \ ε ∂h = 0 , (2.10c) u h = Π u , (2.10d)for all ( r , w, µ ) ∈ V h × W h × M h and n = 1 , , . . . , N . In (2.10), f n = f ( t n , · ), the numerical tracefor the flux on ∂ T h is defined by (cid:98) q nh · n = q nh · n + τ ( u nh − (cid:98) u nh ) , (2.11)and the postprocessed approximate solution u n(cid:63)h is determined on each element K by solving( ∇ u n(cid:63)h , ∇ z h ) K = − ( q nh , ∇ z h ) K , (2.12a)( u n(cid:63)h , w h ) K = ( u nh , w h ) K , (2.12b)for all ( z h , w h ) ∈ [ P k +1 ( K )] ⊥ × P ( K ).As is discussed below, the Interpolatory HDG k method takes great advantage of nodal basisfunctions; however, the postprocessing (2.12) uses an orthogonal complement space, which compli-cates the implementation. To avoid this, on each element K , we introduce a Lagrange multiplier η nh ∈ P ( K ) such that ( ∇ u n(cid:63)h , ∇ z h ) K + ( η nh , z h ) K = − ( q nh , ∇ z h ) K , (2.13a)( u n(cid:63)h , w h ) K = ( u nh , w h ) K , (2.13b)holds for all ( z h , w h ) ∈ P k +1 ( K ) × P ( K ). Remark . In this work, we used w h ∈ P (cid:96) ( K ) with (cid:96) = 0 in (2.13). Actually, (cid:96) = 0 , , . . . , k − (cid:96) = k . 4uperconvergent interpolatory HDG methods for reaction diffusion IAssume V h = span { ϕ i } N i =1 , W h = span { φ i } N i =1 , Z h = span { χ i } N i =1 , and M h = span { ψ i } N i =1 .Then q nh = N (cid:88) j =1 α nj ϕ j , u nh = N (cid:88) j =1 β nj φ j ,u n(cid:63)h = N (cid:88) j =1 γ nj χ j , (cid:98) u nh = N (cid:88) j =1 ζ nj ψ j . (2.14)Also, define the following matrices A = [( ∇ χ j , ∇ χ i ) T h ] , A = [( ϕ j , ∇ χ i ) T h ] , A = [( ϕ j , ϕ i ) T h ] ,A = [( φ j , ∇ · ϕ i ) T h ] , A = [ (cid:104) ψ j , ϕ i · n (cid:105) ∂ T h ] , A = [ (cid:104) τ φ j , φ i (cid:105) ∂ T h ] ,A = [ (cid:104) τ ψ j , ϕ i (cid:105) ∂ T h ] , A = [ (cid:104) τ ψ j , ψ i (cid:105) ∂ T h ] , A = [( χ j , φ i ) T h ] ,M = [( φ j , φ i ) T h ] , and vectors b = [( χ j , T h ] , b = [( φ j , T h ] , b n = [( f n , φ i ) T h ] . Since V h , W h , and Z h are discontinuous finite element spaces, many of the matrices are blockdiagonal with small blocks.Substitute (2.14) into the postprocessing equation (2.13) and use the corresponding test func-tions to test (2.13) on each element K ∈ T h . This gives the following local postprocessing equation (cid:20) A k ( b k ) T b k (cid:21)(cid:20) γ nk η nk (cid:21) = (cid:20) − A k b k (cid:21)(cid:20) α nk β nk (cid:21) , were A k is the k th block of the matrix A , and A k , b k , and b k are defined similarly. That is, (cid:20) γ nk η nk (cid:21) = (cid:20) A k ( b k ) T b k (cid:21) − (cid:20) − A k b k (cid:21)(cid:20) α nk β nk (cid:21) = (cid:20) B k B k B k B k (cid:21)(cid:20) α nk β nk (cid:21) , (2.15)i.e., γ nk = B k α nk + B k β nk . Let B and B be the block diagonal matrices with k th blocks B k and B k , respectively.As in [8], once we test (2.10b) using w = φ i we can express the Interpolatory HDG k nonlinearterm by the matrix-vector product[( I h F ( u n(cid:63)h ) , φ i ) T h ] = A F ( γ n )= A F ( B α n + B β n ) , where F is defined by F ( γ n ) = [ F ( γ n ) , F ( γ n ) , . . . , F ( γ nN )] T . (2.16)5. Chen, B. Cockburn, J. Singler, and Y. ZhangThen the system (2.10) can be rewritten as A − A A A T ∆ t − M + A − A A T A T − A (cid:124) (cid:123)(cid:122) (cid:125) K α n β n ζ n (cid:124) (cid:123)(cid:122) (cid:125) x n + A F ( B α n + B β n )0 (cid:124) (cid:123)(cid:122) (cid:125) F ( x n ) = b n + ∆ t − M β n − (cid:124) (cid:123)(cid:122) (cid:125) b n , (2.17)i.e., K x n + F ( x n ) = b n . (2.18)To apply Newton’s method to solve the nonlinear equations (2.18), define G : R N + N + N → R N + N + N by G ( x n ) = K x n + F ( x n ) − b n . (2.19)At each time step t n for 1 ≤ n ≤ N , given an initial guess x (0) n , Newton’s method yields x ( m ) n = x ( m − n − (cid:104) G (cid:48) ( x ( m − n ) (cid:105) − G ( x ( m − n ) , m = 1 , , , . . . (2.20)where the Jacobian matrix G (cid:48) ( x ( m − n ) is given by G (cid:48) ( x ( m − n ) = K + F (cid:48) ( x ( m − n ) . (2.21)Similar to our earlier work [8] on Interpolatory HDG, the term F (cid:48) ( x ( m − n ) is easily computed by F (cid:48) ( x ( m − n ) = A n, ( m )10 A n, ( m )11
00 0 0 , where A n, ( m )10 and A n, ( m )11 can be efficiently computed using sparse matrix operations by A n, ( m )10 = A diag( F (cid:48) ( B α n, ( m − + B β n, ( m − )) B ,A n, ( m )11 = A diag( F (cid:48) ( B α n, ( m − + B β n, ( m − )) B . Therefore, equation (2.20) can be rewritten as A − A A A T + A n, ( m )10 ∆ t − M + A + A n, ( m )11 − A A T A T − A α n, ( m ) β n, ( m ) ζ n, ( m ) = (cid:101) b , (2.22)where (cid:101) b = G (cid:48) ( x ( m − n ) x ( m − n − G ( x ( m − n ) . (2.23)This equation can be solved by locally eliminating the unknowns α n, ( m ) and β n, ( m ) ; see [8] fordetails. Remark . In this new Interpolatory HDG k formulation, we only need to assemble the HDGmatrices and the HDG postprocessing matrices B and B once before the time integration.Hence, we keep all the advantages from our earlier work [8]: the new approach eliminates thecomputational cost of matrix reassembly and gives simple explicit expressions for the nonlinearterm and Jacobian matrix, which leads to a simple unified implementation for a variety of nonlinearPDEs. 6uperconvergent interpolatory HDG methods for reaction diffusion I In this section, we give a rigorous error analysis for the semidiscrete Interpolatory HDG k method.Below, we state our assumptions and briefly outline the main results. Then we provide an overviewof the projections required for the analysis in Section 3.1. The proofs of the main results follow. Wefirst assume in Section 3.2 that the nonlinearity satisfies a global Lipschitz condition. Finally, inSection 3.3 we extend the results to locally Lipschitz nonlinearities; however, we assume the meshis quasi-uniform and h is sufficiently small for this case.We use the standard notation W m,p (Ω) for Sobolev spaces on Ω with norm (cid:107) · (cid:107) m,p, Ω andseminorm | · | m,p, Ω . We also write H m (Ω) instead of W m, (Ω), and we omit the index p in thecorresponding norms and seminorms.Throughout, we assume the solution of the PDE (1.1) exists and is unique for t ∈ [0 , T ], thefunction F , the problem data, and the solution of the PDE are smooth enough, and the semidiscreteInterpolatory HDG k equations (2.5) have a unique solution on [0 , T ]. Furthermore, we assume themesh is uniformly shape regular, h ≤
1, and the projection Π used for the initial condition in (2.5d)is Π = Π W , where Π W is defined below in Section 3.1.We also make the following regularity assumption on the dual problem: there exists a constant C such that for any Θ ∈ L (Ω), the solution ( Φ , Ψ) of the dual problem Φ + ∇ Ψ = 0 in Ω , ∇ · Φ = Θ in Ω , Ψ = 0 on ∂ Ω , (3.1)satisfies ( Φ , Ψ) ∈ [ H (Ω)] d × H (Ω) and (cid:107) Φ (cid:107) H (Ω) + (cid:107) Ψ (cid:107) H (Ω) ≤ C (cid:107) Θ (cid:107) L (Ω) . (3.2)This assumption is satisfied if Ω is convex.We show for all 0 ≤ t ≤ T the solution ( q h , u h , u (cid:63)h ) of the semidiscrete Interpolatory HDG k equations (2.5) satisfies (cid:107) q ( t ) − q h ( t ) (cid:107) T h ≤ Ch k +1 , (cid:107) u ( t ) − u h ( t ) (cid:107) T h ≤ Ch k +1 , (cid:107) u ( t ) − u (cid:63)h ( t ) (cid:107) T h ≤ Ch k +1+min { k, } . In our error estimates, the constants C can vary from line to line and may depend on the exactsolution and the final time T . As in the linear case [3], superconvergence is only obtained for k ≥ Remark . In [3], the L ∞ ( L ) error for u − u ∗ h superconverges at a rate of √ log κ h k +2 , where κ depends on the mesh and the term √ log κ grows very slowly as h tends to zero. The term √ log κ results from the parabolic duality argument used in [3]. It appears this parabolic duality argumentis not applicable to Interpolatory HDG. Therefore, in this work we use a duality argument basedon Wheeler’s work [37] and avoid the term √ log κ in our error estimates; however, we require thesolution has higher regularity than the regularity needed in [3] for the linear case. We first introduce the HDG k projection operator Π h ( q , u ) := ( Π V q , Π W u ) defined in [15], where Π V q and Π W u denote components of the projection of q and u into V h and W h , respectively. Foreach element K ∈ T h , the projection is determined by the equations( Π V q , r ) K = ( q , r ) K , ∀ r ∈ [ P k − ( K )] d , (3.3a)(Π W u, w ) K = ( u, w ) K , ∀ w ∈ P k − ( K ) , (3.3b) (cid:104) Π V q · n + τ Π W u, µ (cid:105) e = (cid:104) q · n + τ u, µ (cid:105) e , ∀ µ ∈ P k ( e ) , (3.3c)7. Chen, B. Cockburn, J. Singler, and Y. Zhangfor all faces e of the simplex K . The approximation properties of the HDG k projection (3.3) aregiven in the following result from [15]: Lemma 3.2.
Suppose k ≥ , τ | ∂K is nonnegative and τ max K := max τ | ∂K > . Then the system (3.3) is uniquely solvable for Π V q and Π W u . Furthermore, there is a constant C independent of K and τ such that (cid:107) Π V q − q (cid:107) K ≤ Ch (cid:96) q +1 K | q | (cid:96) q +1 ,K + Ch (cid:96) u +1 K τ ∗ K | u | (cid:96) u +1 ,K , (3.4a) (cid:107) Π W u − u (cid:107) K ≤ Ch (cid:96) u +1 K | u | (cid:96) u +1 ,K + C h (cid:96) q +1 K τ max K |∇ · q | (cid:96) q ,K (3.4b) for (cid:96) q , (cid:96) u in [0 , k ] . Here τ ∗ K := max τ | ∂K \ e ∗ , where e ∗ is a face of K at which τ | ∂K is maximum. Next, for each simplex K in T h and each boundary face e of K , let Π (cid:96) (for any (cid:96) ≥
0) and P M denote the standard L orthogonal projection operators Π (cid:96) : L ( K ) → P (cid:96) ( K ) and P M : L ( e ) →P k ( e ) satisfying (Π (cid:96) u, v h ) K = ( u, v h ) K , ∀ v h ∈ P (cid:96) ( K ) , (3.5a) (cid:104) P M u, (cid:98) v h (cid:105) e = (cid:104) u, (cid:98) v h (cid:105) e , ∀ (cid:98) v h ∈ P k ( e ) . (3.5b)The following error estimates for the L projections and the elementwise interpolation operator I h from Section 2 are standard and can be found in [1]: Lemma 3.3.
Suppose k, (cid:96) ≥ . There exists a constant C independent of K ∈ T h such that (cid:107) w − I h w (cid:107) K + h K (cid:107)∇ ( w − I h w ) (cid:107) K ≤ Ch k +2 (cid:107) w (cid:107) k +2 ,K , ∀ w ∈ C ( ¯ K ) ∩ H k +2 ( K ) , (3.6a) (cid:107) w − Π (cid:96) w (cid:107) K ≤ Ch (cid:96) +1 (cid:107) w (cid:107) (cid:96) +1 ,K , ∀ w ∈ H (cid:96) +1 ( K ) , (3.6b) (cid:107) w − P M w (cid:107) ∂K ≤ Ch k +1 / (cid:107) w (cid:107) k +1 ,K ∀ w ∈ H k +1 ( K ) . (3.6c) In this section, we assume the nonlinearity F is globally Lipschitz: Assumption 3.4.
There is a constant
L > | F ( u ) − F ( v ) | R ≤ L | u − v | R for all u, v ∈ R .We remove this restriction in the next section. Our proof relies on techniques used in [3, 8]. Wesplit the proof of the main result into several steps.To begin, we first rewrite the semidiscrete interpolatory HDG equations (2.5). First, subtract(2.5c) from (2.5b) and integrate by parts to give the following formulation: Lemma 3.5.
The Interpolatory HDG k method finds ( q h , u h , (cid:98) u h ) ∈ V h × W h × M h satisfying ( q h , r h ) T h − ( u h , ∇ · r h ) T h + (cid:104) (cid:98) u h , r h · n (cid:105) ∂ T h = 0 , (3.7a)( ∂ t u h , v h ) T h + ( I h F ( u (cid:63)h ) , v h ) T h + ( ∇ · q h , v h ) T h − (cid:104) q h · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( u h − (cid:98) u h ) , v h − (cid:98) v h ) (cid:105) ∂ T h = ( f, v h ) T h , (3.7b) u h (0) = Π W u , (3.7c) for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . k operator B : B ( q h , u h , (cid:98) u h ; r h , v h , (cid:98) v h )= ( q h , r h ) T h − ( u h , ∇ · r h ) T h + (cid:104) (cid:98) u h , r h · n (cid:105) ∂ T h + ( ∇ · q h , v h ) T h − (cid:104) q h · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( u h − (cid:98) u h ) , v h − (cid:98) v h ) (cid:105) ∂ T h . (3.8)This allows us to rewrite the semidiscrete Interpolatory HDG k formulation (3.7) as follows: find( q h , u h , (cid:98) u h ) ∈ V h × W h × M h such that( ∂ t u h , v h ) T h + B ( q h , u h , (cid:98) u h ; r h , v h , (cid:98) v h ) + ( I h F ( u (cid:63)h ) , v h ) T h = ( f, v h ) , (3.9a) u h (0) = Π W u . (3.9b)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . For ε q h = Π V q − q h , ε uh = Π W u − u h , and ε (cid:98) uh = P M u − (cid:98) u h , we have ( ∂ t ε uh , v h ) T h + B ( ε q h , ε uh , ε (cid:98) uh ; r h , w h , (cid:98) v h ) + ( F ( u ) − I h F ( u (cid:63)h ) , v h ) T h = ( Π V q − q , r h ) T h + (Π W u t − u t , v h ) T h , (3.10a) ε uh | t =0 = 0 , (3.10b) for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h .Proof. By the definition of the operator B in (3.8), we have B ( Π V q , Π W u, P M u, r h , v h , (cid:98) v h )= ( Π V q , r h ) T h − (Π W u, ∇ · r h ) T h + (cid:104) P M u, r h · n (cid:105) ∂ T h + ( ∇ · Π V q , v h ) T h − (cid:104) Π V q · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π W u − u ) , v h − (cid:98) v h (cid:105) ∂ T h = ( Π V q − q , r h ) T h + ( q , r h ) T h − ( u, ∇ · r h ) T h + (cid:104) u, r h · n (cid:105) ∂ T h + ( ∇ · q , v h ) T h + ( ∇ · ( Π V q − q ) , v h ) T h − (cid:104) ( Π V q − q ) · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π W u − u ) , v h − (cid:98) v h (cid:105) ∂ T h = ( Π V q − q , r h ) T h + ( f − F ( u ) + ∂ t u, v h ) T h , where we used the HDG k projection (3.3) and the L projection P M (3.5b). Use (3.9) and subtractto obtain the result. ε uh in L ∞ ( L ) by an energy argumentLemma 3.7. For any t ∈ [0 , T ] , we have (cid:107) Π k +1 u − u (cid:63)h (cid:107) T h ≤ C ( (cid:107) ε uh (cid:107) T h + (cid:107) u − I h u (cid:107) K + δ k (cid:107) Π W u − u (cid:107) T h )+ Ch ( (cid:107) ε q h (cid:107) T h + (cid:107) q − Π V q (cid:107) T h + (cid:107)∇ ( u − I h u ) (cid:107) T h ) , where δ k denotes the Kronecker delta symbol so that δ k = 1 for k = 0 and δ k = 0 for k ≥ .
9. Chen, B. Cockburn, J. Singler, and Y. Zhang
Proof.
We begin the proof with the case k ≥
1. The proof is very similar to a proof in [33], but weinclude it for completeness. By the properties of Π W and Π k +1 , we obtain(Π W u, w ) K = ( u, w ) K , for all w ∈ P ( K ) , (Π k +1 u, w ) K = ( u, w ) K , for all w ∈ P ( K ) . Hence, for all w ∈ P ( K ), we have (Π W u − Π k +1 u, w ) K = 0 . Let e h = u (cid:63)h − u h + Π W u − Π k +1 u . Using the postprocessing equation (2.7), q = −∇ u , and aninverse inequality gives (cid:107)∇ e h (cid:107) K = ( ∇ ( u (cid:63)h − u h ) , ∇ e h ) K + ( ∇ (Π W u − Π k +1 u ) , ∇ e h ) K = ( −∇ u h − q h , ∇ e h ) K + ( ∇ (Π W u − Π k +1 u ) , ∇ e h ) K = ( ∇ (Π W u − u h ) − ( q h − Π V q ) + ( q − Π V q ) + ∇ ( u − Π k +1 u ) , ∇ e h ) K ≤ C ( h − K (cid:107) Π W u − u h (cid:107) K + (cid:107) ε q h (cid:107) K + (cid:107) q − Π V q (cid:107) K + (cid:107)∇ ( u − Π k +1 u ) (cid:107) K ) (cid:107)∇ e h (cid:107) K . (3.11)Since ( e h , K = 0, apply the Poincar´e inequality and the above estimate (3.11) to give (cid:107) e h (cid:107) K ≤ Ch K (cid:107)∇ e h (cid:107) K ≤ C (cid:107) ε uh (cid:107) K + Ch K ( (cid:107) ε q h (cid:107) K + (cid:107) q − Π V q (cid:107) K + (cid:107)∇ ( u − Π k +1 u ) (cid:107) K ) . Next, estimate the last term using an inverse inequality: h K (cid:107)∇ ( u − Π k +1 u ) (cid:107) K ≤ h K (cid:107)∇ ( u − I h u ) (cid:107) K + h K (cid:107)∇ ( I h u − Π k +1 u ) (cid:107) K ≤ h (cid:107)∇ ( u − I h u ) (cid:107) K + (cid:107)I h u − Π k +1 u (cid:107) K ≤ h (cid:107)∇ ( u − I h u ) (cid:107) K + (cid:107) u − I h u (cid:107) K . This implies (cid:107) e h (cid:107) T h ≤ C ( (cid:107) ε uh (cid:107) T h + (cid:107) u − I h u (cid:107) T h ) + Ch ( (cid:107) ε q h (cid:107) T h + (cid:107) q − Π V q (cid:107) T h + (cid:107)∇ ( u − I h u ) (cid:107) T h ) . Hence, we have (cid:107) Π k +1 u − u (cid:63)h (cid:107) T h ≤ (cid:107) Π k +1 u − Π W u − u (cid:63)h + u h (cid:107) T h + (cid:107) Π W u − u h (cid:107) T h ≤ C ( (cid:107) ε uh (cid:107) T h + (cid:107) u − I h u (cid:107) T h ) + Ch ( (cid:107) ε q h (cid:107) T h + (cid:107) q − Π V q (cid:107) T h + (cid:107)∇ ( u − I h u ) (cid:107) T h ) . This completes the proof for the case k ≥ k = 0, we follow the above steps in the proof of the k ≥ W with Π k to obtain (cid:107) Π k +1 u − u (cid:63)h (cid:107) T h ≤ (cid:107) Π k +1 u − Π k u − u (cid:63)h + u h (cid:107) T h + (cid:107) Π k u − u h (cid:107) T h ≤ C ( (cid:107) Π k u − u h (cid:107) T h + (cid:107) u − I h u (cid:107) T h )+ Ch ( (cid:107) ε q h (cid:107) T h + (cid:107) q − Π V q (cid:107) T h + (cid:107)∇ ( u − I h u ) (cid:107) T h ) ≤ C ( (cid:107) ε uh (cid:107) T h + (cid:107) Π k u − u (cid:107) T h + (cid:107) Π W u − u (cid:107) T h + (cid:107) u − I h u (cid:107) T h )+ Ch ( (cid:107) ε q h (cid:107) T h + (cid:107) q − Π V q (cid:107) T h + (cid:107)∇ ( u − I h u ) (cid:107) T h ) . The optimality of the L projection gives (cid:107) Π k u − u (cid:107) T h ≤ (cid:107) Π W u − u (cid:107) T h , and this completes theproof. 10uperconvergent interpolatory HDG methods for reaction diffusion ITo bound the error in the nonlinear term, we split F ( u ) − I h F ( u (cid:63)h ) as F ( u ) − I h F ( u (cid:63)h ) = F ( u ) − I h F ( u ) + I h F ( u ) − I h F (Π k +1 u ) + I h F (Π k +1 u ) − I h F ( u (cid:63)h )=: R + R + R . A bound for the first term R follows directly from the standard FE interpolation error estimate(3.6a) in Lemma 3.3 due to the smoothness assumption for the function F . Error bounds for R and R are given in the following result: Lemma 3.8.
We have (cid:107)I h F ( u ) − I h F (Π k +1 u ) (cid:107) T h ≤ C (cid:107) u − I h u (cid:107) T h , (cid:107)I h F (Π k +1 u ) − I h F ( u (cid:63)h ) (cid:107) T h ≤ C (cid:107) Π k +1 u − u (cid:63)h (cid:107) T h . The proofs of the estimates in Lemma 3.8 are similar to proofs in [8] and also use (cid:107) Π k +1 u − u (cid:107) T h ≤(cid:107) u − I h u (cid:107) T h . We omit the details. Lemma 3.9.
We have the estimate (cid:107) ε uh ( t ) (cid:107) T h + (cid:90) t (cid:104) (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h (cid:105) dt ≤ C (cid:90) t H , where H = (cid:107) Π V q − q (cid:107) T h + (cid:107) Π W u t − u t (cid:107) T h + δ k (cid:107) Π W u − u (cid:107) T h + (cid:107) F ( u ) − I h F ( u ) (cid:107) T h + (cid:107) u − I h u (cid:107) T h + h (cid:107)∇ ( u − I h u ) (cid:107) T h . (3.12) Proof.
Take ( r h , v h , (cid:98) v h ) = ( ε q h , ε uh , ε (cid:98) uh ) in the error equation (3.10) to give12 ddt (cid:107) ε uh (cid:107) T h + (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h = ( Π V q − q , ε q h ) T h + (Π W u t − u t , ε uh ) T h − ( F ( u ) − I h F ( u (cid:63)h ) , ε uh ) T h . (3.13)Apply the Cauchy-Schwarz inequality to each term of the right-hand side of the above identity anduse h ≤
1, Lemma 3.7, and Lemma 3.8 to get ddt (cid:107) ε uh (cid:107) T h + (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h ≤ C H + C (cid:107) ε uh (cid:107) T h . Gronwall’s inequality, ε uh (0) = 0, and e Ct ≤ e CT give the result. ε qh in L ∞ ( L ) by an energy argumentLemma 3.10. We have (cid:107) ε q h ( t ) (cid:107) T h + (cid:107)√ τ ( ε uh ( t ) − ε (cid:98) uh ( t )) (cid:107) ∂ T h ≤ C (cid:18) (cid:107) ( Π V q − q )(0) (cid:107) T h + (cid:90) t G (cid:19) , where G = (cid:107) Π V q − q (cid:107) T h + (cid:107) Π W u t − u t (cid:107) T h + (cid:107) Π V q t − q t (cid:107) T h + δ k (cid:107) Π W u − u (cid:107) T h + (cid:107) F ( u ) − I h F ( u ) (cid:107) T h + (cid:107) u − I h u (cid:107) T h + h (cid:107)∇ ( u − I h u ) (cid:107) T h . (3.14)11. Chen, B. Cockburn, J. Singler, and Y. Zhang Proof.
Take ( r h , v h , (cid:98) v h ) = ( r h , ,
0) in the error equation (3.10) and differentiate the result withrespect to time. Also take ( r h , v h , (cid:98) v h ) = (0 , v h , (cid:98) v h ) in (3.10) to get( ∂ t ε q h , r h ) T h − ( ∂ t ε uh , ∇ · r h ) T h + (cid:104) ∂ t ε (cid:98) uh , r h · n (cid:105) ∂ T h = ( Π V q t − q t , r h ) T h , (3.15a)( ∂ t ε uh , v h ) T h + ( ∇ · ε q h , v h ) T h − (cid:104) ε q h · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , v h − (cid:98) v h (cid:105) ∂ T h + ( F ( u ) − I h F ( u (cid:63)h ) , v h ) T h = (Π W u t − u t , v h ) T h , (3.15b) ε uh | t =0 = 0 , (3.15c)for all ( r h , w h , (cid:98) v h ) ∈ V h × W h × M h .Next, take r h = ε q h in (3.15a), v h = ∂ t ε uh in (3.15b), and (cid:98) v h = ∂ t ε (cid:98) uh in (3.15b) to obtain (cid:107) ∂ t ε uh (cid:107) T h + ( ∂ t ε q h , ε q h ) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ∂ t ε uh − ∂ t ε (cid:98) uh (cid:105) ∂ T h = ( Π V q t − q t , ε q h ) T h + (Π W u t − u t , ∂ t ε uh ) T h − ( F ( u ) − I h F ( u (cid:63)h ) , ∂ t ε uh ) T h . Integrating in time gives12 [ (cid:107) ε q h ( t ) (cid:107) T h + (cid:107)√ τ ( ε uh ( t ) − ε (cid:98) uh ( t )) (cid:107) ∂ T h ] + (cid:90) t (cid:107) ∂ t ε uh (cid:107) T h = 12 [ (cid:107) ε q h (0) (cid:107) T h + (cid:107)√ τ ( ε uh (0) − ε (cid:98) uh (0)) (cid:107) ∂ T h ] + (cid:90) t ( Π V q t − q t , ε q h ) T h + (cid:90) t (Π W u t − u t , ∂ t ε uh ) T h − (cid:90) t ( F ( u ) − I h F ( u (cid:63)h ) , ∂ t ε uh ) T h . Use the Cauchy-Schwarz inequality, h ≤
1, Lemma 3.7, and Lemma 3.8 to get (cid:107) ε q h ( t ) (cid:107) T h + (cid:107)√ τ ( ε uh ( t ) − ε (cid:98) uh ( t )) (cid:107) ∂ T h ≤ (cid:107) ε q h (0) (cid:107) T h + (cid:107)√ τ ( ε uh (0) − ε (cid:98) uh (0)) (cid:107) ∂ T h + C (cid:90) t (cid:0) G + (cid:107) ε uh (cid:107) T h + (cid:107) ε q h (cid:107) T h (cid:1) . Apply the integral Gronwall’s inequality to obtain (cid:107) ε q h ( t ) (cid:107) T h + (cid:107)√ τ ( ε uh ( t ) − ε (cid:98) uh ( t )) (cid:107) ∂ T h ≤ C (cid:18) (cid:107) ε q h (0) (cid:107) T h + (cid:107)√ τ ( ε uh (0) − ε (cid:98) uh (0)) (cid:107) ∂ T h + (cid:90) t G + (cid:90) t (cid:107) ε uh (cid:107) T h (cid:19) . Next, let t = 0 in (3.13) and use ε uh (0) = 0 to get (cid:107) ε q h (0) (cid:107) T h + (cid:107)√ τ ( ε uh − ε (cid:98) uh )(0) (cid:107) ∂ T h = (( Π V q − q )(0) , ε q h (0)) T h . Therefore, (cid:107) ε q h (0) (cid:107) T h + (cid:107)√ τ ( ε uh − ε (cid:98) uh )(0) (cid:107) ∂ T h ≤ (cid:107) ( Π V q − q )(0) (cid:107) T h , and the estimate for (cid:107) ε uh (cid:107) T h in Lemma 3.9 completes the proof.12uperconvergent interpolatory HDG methods for reaction diffusion I ε uh in L ∞ ( L ) by a duality argument To get a superconvergent rate for (cid:107) ε uh (cid:107) T h , we adopt a duality argument from Wheeler [37]. In thatwork, an elliptic projection is used and it commutes with the time derivative. It is easy to checkthat the operator Π W defined in (3.3) commutes with the time derivative, i.e, ∂ t Π W u = Π W u t .For any t ∈ [0 , T ], let ( q h , u h , (cid:98) u h ) ∈ V h × W h × M h be the solution of the following steady stateproblem B ( q h , u h , (cid:98) u h , r h , v h , (cid:98) v h ) = ( f − Π W u t − F ( u ) , v h ) T h , (3.16)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h .The following estimates are proved in Section 6. Lemma 3.11.
For any t ∈ [0 , T ] , we have (cid:107) Π V q − q h (cid:107) T h ≤ C ( (cid:107) q − Π V q (cid:107) T h + (cid:107) u t − Π W u t (cid:107) T h ) , (3.17a) (cid:107) Π W u − u h (cid:107) T h ≤ Ch min { k, } ( (cid:107) q − Π V q (cid:107) T h + (cid:107) u t − Π W u t (cid:107) T h ) , (3.17b) (cid:107) ∂ t (Π W u − u h ) (cid:107) T h ≤ Ch min { k, } ( (cid:107) q t − Π V q t (cid:107) T h + (cid:107) u tt − Π W u tt (cid:107) T h ) . (3.17c) Lemma 3.12.
Let e q h = q h − q h , e uh = u h − u h , and e (cid:98) uh = (cid:98) u h − (cid:98) u h . Then for any t ∈ [0 , T ] we have (cid:107) e uh ( t ) (cid:107) T h ≤ (cid:107) Π W u − u h (0) (cid:107) T h + C (cid:90) t (cid:0) (cid:107) ∂ t (Π W u − u h ) (cid:107) T h + h (cid:107) ( Π V q − q )(0) (cid:107) T h + K (cid:1) , where K = h ( (cid:107) Π V q − q (cid:107) T h + (cid:107) Π W u t − u t (cid:107) T h + (cid:107) Π V q t − q t (cid:107) T h )+ (cid:107) F ( u ) − I h F ( u ) (cid:107) T h + (cid:107) u − I h u (cid:107) T h + h (cid:107)∇ ( u − I h u ) (cid:107) T h + δ k (cid:107) Π W u − u (cid:107) T h . (3.18) Proof.
By the definition of the operator B in (3.8), we have( ∂ t e uh , v h ) T h + B ( e q h , e uh , e (cid:98) uh , r h , v h , (cid:98) v h )= ( ∂ t u h , v h ) T h + B ( q h , u h , (cid:98) u h , r h , v h , (cid:98) v h ) − ( ∂ t u h , v h ) T h − B ( q h , u h , (cid:98) u h , r h , v h , (cid:98) v h )= ( f, v h ) T h − ( I h F ( u (cid:63)h ) , v h ) T h − ( ∂ t u h , v h ) T h − ( f − Π W u t − F ( u ) , v h ) T h = ( ∂ t (Π W u − u h ) , v h ) T h + ( F ( u ) − I h F ( u (cid:63)h ) , v h ) T h . Take ( r h , v h , (cid:98) v h ) = ( e q h , e uh , e (cid:98) uh ) and use Lemma 3.7, Lemma 3.8, Lemma 3.10, the bound (cid:107) ε uh (cid:107) = (cid:107) Π W u − u h − e uh (cid:107) ≤ (cid:107) Π W u − u h (cid:107) + (cid:107) e uh (cid:107) , and also Lemma 3.11 to give12 ddt (cid:107) e uh (cid:107) T h + (cid:107) e q h (cid:107) T h + (cid:104) τ ( e uh − e (cid:98) uh ) , e uh − e (cid:98) uh (cid:105) ∂ T h ≤ C (cid:0) K + (cid:107) ∂ t (Π W u − u h ) (cid:107) T h + h (cid:107) ( Π V q − q )(0) (cid:107) T h + C (cid:107) e uh (cid:107) T h (cid:1) . Integration from 0 to t gives (cid:107) e uh ( t ) (cid:107) T h + (cid:90) t (cid:104) (cid:107) e q h (cid:107) T h + (cid:104) τ ( e uh − e (cid:98) uh ) , e uh − e (cid:98) uh (cid:105) ∂ T h (cid:105) ≤ (cid:107) e uh (0) (cid:107) T h + C (cid:90) t ( (cid:107) ∂ t (Π W u − u h ) (cid:107) T h + h (cid:107) ( Π V q − q )(0) (cid:107) T h + K ) + C (cid:90) t (cid:107) e uh (cid:107) T h .
13. Chen, B. Cockburn, J. Singler, and Y. ZhangBy Gronwall’s inequality and e uh (0) = u h (0) − u h (0) = Π W u − u h (0), we have (cid:107) e uh ( t ) (cid:107) T h ≤ (cid:107) Π W u − u h (0) (cid:107) T h + C (cid:90) t ( (cid:107) ∂ t (Π W u − u h ) (cid:107) T h + h (cid:107) ( Π V q − q )(0) (cid:107) T h + K ) . A combination of Lemma 3.11 and Lemma 3.12 gives the following lemma:
Lemma 3.13.
For any t ∈ [0 , T ] , we have (cid:107) ε uh ( t ) (cid:107) T h ≤ Ch min { k, } ( (cid:107) q (0) − Π V q (0) (cid:107) T h + (cid:107) u t (0) − Π W u t (0) (cid:107) T h ) + C (cid:90) t L , where L = h ( (cid:107) Π V q − q (cid:107) T h + (cid:107) Π W u t − u t (cid:107) T h ) + h min { k, } ( (cid:107) u tt − Π W u tt (cid:107) T h + (cid:107) Π V q t − q t (cid:107) T h )+ (cid:107) F ( u ) − I h F ( u ) (cid:107) T h + (cid:107) u − I h u (cid:107) T h + h (cid:107)∇ ( u − Π k +1 u ) (cid:107) T h + δ k (cid:107) Π W u − u (cid:107) T h . (3.19)Using u − u h = ( u − Π W u ) + ε uh , q − q h = ( q − Π V q ) + ε q h , u − u (cid:63)h = ( u − Π k +1 u ) + (Π k +1 u − u (cid:63)h ),and the triangle inequality gives the main result: Theorem 3.14.
If the nonlinearity F satisfies the global Lipschitz condition in Assumption 3.4 andthe assumptions at the beginning of Section 3 hold, then for all ≤ t ≤ T the solution ( q h , u h , u (cid:63)h ) of the semidiscrete Interpolatory HDG k equations satisfy (cid:107) q ( t ) − q h ( t ) (cid:107) T h ≤ (cid:107) q ( t ) − Π V q ( t ) (cid:107) T h + C (cid:107) ( Π V q − q )(0) (cid:107) T h + C (cid:90) t G , (cid:107) u ( t ) − u h ( t ) (cid:107) T h ≤ (cid:107) u ( t ) − Π W u ( t ) (cid:107) T h + C (cid:90) t H , (cid:107) u ( t ) − u (cid:63)h ( t ) (cid:107) T h ≤ Ch min { k, } ( (cid:107) q (0) − Π V q (0) (cid:107) T h + (cid:107) u t (0) − Π W u t (0) (cid:107) T h )+ (cid:107) u ( t ) − Π k +1 u ( t ) (cid:107) T h + C (cid:90) t L , where G , H and L are defined in (3.14) , (3.12) and (3.19) , respectively. By Lemma 3.2, Lemma 3.3, and Theorem 3.14, we obtain convergence rates for smooth solutions.
Corollary 3.15.
If, in addition, u , q , and F ( u ) are sufficiently smooth for t ∈ [0 , T ] , then for all ≤ t ≤ T the solution ( q h , u h , u (cid:63)h ) of the semidiscrete Interpolatory HDG k equations satisfy (cid:107) q ( t ) − q h ( t ) (cid:107) T h ≤ Ch k +1 , (cid:107) u ( t ) − u h ( t ) (cid:107) T h ≤ Ch k +1 , (cid:107) u ( t ) − u (cid:63)h ( t ) (cid:107) T h ≤ Ch k +1+min { k, } . In applications, the nonlinearity F might not satisfy the global Lipschitz condition, see Assump-tion 3.4. Instead, let M = max {| u ( t, x ) | , x ∈ Ω , t ∈ [0 , T ] } + 1 . (3.20)In this section, we assume the mesh is quasi-uniform, the polynomial degree satisfies k ≥
1, andthe nonlinearity F satisfies the following local Lipschitz condition:14uperconvergent interpolatory HDG methods for reaction diffusion I Assumption 3.16.
There is a constant L ( M ) > | F ( u ) − F ( v ) | R ≤ L ( M ) | u − v | R for all u, v ∈ [ − M, M ].Our proof relies on techniques used in [36]. Below, we use the notation ρ u (cid:63) h = Π k +1 u − u (cid:63)h . Lemma 3.17. If h is small enough and k ≥ , then there exists t ∗ h ∈ (0 , T ] such that Lemma 3.10and Lemma 3.13 hold for all t ∈ [0 , t ∗ h ] .Proof. Take ( r h , v h , (cid:98) v h ) = ( ε q h , ε uh , ε (cid:98) uh ) in (3.10) to give12 ddt (cid:107) ε uh (cid:107) T h + (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h = ( Π V q − q , ε q h ) T h − ( F ( u ) − I h F ( u (cid:63)h ) , ε uh ) T h . (3.21)Take t = 0 and use the fact ε uh (0) = 0 to obtain (cid:107) ε q h (0) (cid:107) T h ≤ C (cid:107) Π V q (0) − q (0) (cid:107) T h . By Lemma 3.7, we have (cid:107) ρ u (cid:63) h (0) (cid:107) T h ≤ C (cid:107) u (0) − I h u (0) (cid:107) T h + Ch ( (cid:107) Π V q (0) − q (0) (cid:107) T h + (cid:107)∇ ( u (0) − I h u (0)) (cid:107) T h ) . The inverse inequality gives (cid:107) ρ u (cid:63) h (0) (cid:107) L ∞ (Ω) ≤ Ch − d (cid:107) u (0) − I h u (0) (cid:107) T h + Ch − d ( (cid:107) Π V q (0) − q (0) (cid:107) T h + (cid:107)∇ ( u (0) − I h u (0)) (cid:107) T h ) . Since the exact solution is smooth at t = 0, we can choose h small enough so that (cid:107) ρ u (cid:63) h (0) (cid:107) L ∞ (Ω) < /
2. Also, since the error equation (3.10) is continuous with respect to the time t , again using aninverse inequality shows that there exists t ∗ h ∈ (0 , T ] such that for all h small enough, (cid:107) ρ u (cid:63) h ( t ) (cid:107) L ∞ (Ω) ≤ / t ∈ [0 , t ∗ h ]. (3.22)For all h sufficiently small we have (cid:107) u ( t ) − Π k +1 u ( t ) (cid:107) L ∞ (Ω) ≤ / t ∈ [0 , t ∗ h ]. (3.23)This implies for all h small enough and all t ∈ [0 , t ∗ h ], (cid:107) Π k +1 u (cid:107) L ∞ ≤ (cid:107) u (cid:107) L ∞ + (cid:107) u − Π k +1 u (cid:107) L ∞ ≤ (cid:107) u (cid:107) L ∞ + 1 / ≤ M, (cid:107) u (cid:63)h (cid:107) L ∞ ≤ (cid:107) Π k +1 u (cid:107) L ∞ + (cid:107) Π k +1 u − u (cid:63)h (cid:107) L ∞ ≤ ( (cid:107) u (cid:107) L ∞ + 1 /
2) + 1 / M. Therefore, u , Π k +1 u , and u (cid:63)h are located in the interval [ − M, M ], where M is defined in (3.20).This allows us to take advantage of the local Lipschitz condition of F ( u ) for all t ∈ [0 , t ∗ h ]. Hence,Lemma 3.10 and Lemma 3.13 hold for all t ∈ [0 , t ∗ h ]. Lemma 3.18.
For h small enough and k ≥ , the conclusions of Lemma 3.10 and Lemma 3.13are true on the whole time interval [0 , T ] .
15. Chen, B. Cockburn, J. Singler, and Y. ZhangTable 1: History of convergence.
Example 1: Errors for q h , u h and u (cid:63)h of Interpolatory HDG k Degree h √ (cid:107) q − q h (cid:107) , Ω (cid:107) u − u h (cid:107) , Ω (cid:107) u − u (cid:63)h (cid:107) , Ω Error Rate Error Rate Error Rate k = 0 2 − − − − − k = 1 2 − − − − − Proof.
Fix h ∗ > h ≤ h ∗ , and assume t ∗ h isthe largest value for which (3.22) is true for all h ≤ h ∗ . Define the set A = { h ∈ [0 , h ∗ ] : t ∗ h (cid:54) = T } .If the result is not true, then A is nonempty, inf { h : h ∈ A} = 0, and also (cid:107) ρ u (cid:63) h ( t ∗ h ) (cid:107) L ∞ (Ω) = 1 / h ∈ A . (3.24)However, by the inverse inequality, k ≥
1, and since Lemma 3.17 is true, we have (cid:107) ρ u (cid:63) h ( t ∗ h ) (cid:107) L ∞ (Ω) ≤ Ch − d (cid:107) ρ u (cid:63) h ( t ∗ h ) (cid:107) T h ≤ Ch − d/ for all h ∈ A .Since C does not depend on h , there exists h ∗ ≤ h ∗ such that (cid:107) ρ u (cid:63) h ( t ∗ h ) (cid:107) L ∞ (Ω) < / h ∈ A such that h ≤ h ∗ . This contradicts (3.24), and therefore t ∗ h = T for all h small enough. Theorem 3.19.
If the nonlinearity F satisfies the local Lipschitz condition in Assumption 3.16,the mesh is quasi-uniform, k ≥ , and the assumptions at the beginning of Section 3 hold, then forall h small enough the conclusions of Theorem 3.14 and Corollary 3.15 are true for all ≤ t ≤ T . In this section, we present two examples to demonstrate the performance of the Interpolatory HDG k method. Example 4.1 (The Allen-Cahn or Chaffee-Infante equation) . We begin with an example withan exact solution in order to illustrate the convergence theory. The domain is the unit squareΩ = [0 , × [0 , ⊂ R , the nonlinear term is F ( u ) = u − u , and the source term f is chosenso that the exact solution is u = sin( t ) sin( πx ) sin( πy ). Backward Euler and Crank-Nicolson areapplied for the time discretization when k = 0 and k = 1, respectively, where k is the degree of thepolynomial. The time step is chosen as ∆ t = h when k = 0 and ∆ t = h when k = 1. We reportthe errors at the final time T = 1 for polynomial degrees k = 0 and k = 1 in Table 1. The observedconvergence rates match the theory. Example 4.2 (The Schnakenberg model) . Next, we consider a more complicated example of areaction diffusion PDE system with zero Neumann boundary conditions that does not satisfy theassumptions for the convergence theory established here. We consider such an example to demon-strate the applicability of the Interpolatory HDG k method to more general problems.16uperconvergent interpolatory HDG methods for reaction diffusion ISpecifically, we consider the Schnakenberg model, which has been used to model the spatialdistribution of a morphogen; see [42] for more details. The Schnakenberg system has the form ∂C a ∂t = D ∇ C a + κ ( a − C a + C a C i ) ,∂C i ∂t = D ∇ C i + κ ( b − C a C i ) , with initial conditions C a ( · ,
0) = a + b + 10 − exp (cid:0) − x − / + ( y − / ) (cid:1) ,C i ( · ,
0) = b ( a + b ) , and homogeneous Neumann boundary conditions. The parameter values are κ = 100, a = 0 . b = 0 . D = 0 .
05, and D = 1. We choose polynomial degree k = 1 and apply Crank-Nicolsonfor the time discretization with time step ∆ t = 0 . , × [0 , { ( x, y ) : ( x − . + ( y − . < . } and we use 7168 elements.Numerical results are shown in Figure 1–Figure 2. Spot patterns form on the square and circulardomains. Our numerical results are very similar to results reported in [42]. In our earlier work [8], we considered an Interpolatory HDG k methods for semilinear parabolicPDEs with a general nonlinearity of the form F ( ∇ u, u ). The interpolatory approach achievesoptimal convergence rates and reduces the computational cost compared to standard HDG sinceall of the HDG matrices are assembled once before the time stepping procedure. However, themethod does not have superconvergence by postprocessing.In this work, we proposed a new superconvergent Interpolatory HDG k method for approximatingthe solution of reaction diffusion PDEs. Unlike our earlier Interpolatory HDG k work [8], thenew method uses a postprocessing u (cid:63)h to evaluate the nonlinear term. This change provides thesuperconvergence, and the new method also keeps all of the computational advantages of using aninterpolatory approach for the nonlinear term. We proved the superconvergence under a globalLipschitz condition for the nonlinearity, and then extended the superconvergence results to a localLipschitz condition assuming the mesh is quasi-uniform.In the second part of this work [5], we again consider reaction diffusion equations and extendthe ideas here to derive other superconvergent interpolatory HDG methods inspired by hybrid high-order methods [9]. However, it is currently not clear whether the present approach can be usedto obtain the superconvergence for semilinear PDEs with a general nonlinearity F ( ∇ u, u ). We arecurrently exploring this issue. Recall the steady state problem (3.16) from Section 3.2.4, which we repeat here for convenience:let ( q h , u h , (cid:98) u h ) ∈ V h × W h × M h be the solution of B ( q h , u h , (cid:98) u h , r h , v h , (cid:98) v h ) = ( f − Π W u t − F ( u ) , v h ) T h , (6.1)17. Chen, B. Cockburn, J. Singler, and Y. ZhangFigure 1: Contour plots of the time evolution of the concentration of the activator C a on the unitsquare. 18uperconvergent interpolatory HDG methods for reaction diffusion IFigure 2: Contour plots of the time evolution of the concentration of the activator C a on thecircular domain. 19. Chen, B. Cockburn, J. Singler, and Y. Zhangfor all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . Since Π W commutes with the time derivative, taking the partialderivative of (6.1) with respect to t shows ( ∂ t q h , ∂ t u h , ∂ t (cid:98) u h ) ∈ V h × W h × M h is the solution of B ( ∂ t q h , ∂ t u h , ∂ t (cid:98) u h , r h , v h , (cid:98) v h ) = ( f t − Π W u tt − F (cid:48) ( u ) u t , v h ) T h , (6.2)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h .The proof of the following lemma is very similar to a proof in [15], hence we omit it here. Lemma 6.1.
For ε q h = Π V q − q h , ε uh = Π W u − u h , and ε (cid:98) uh = P M u − (cid:98) u h , we have B ( ε q h , ε uh , ε (cid:98) uh ; r h , w h , (cid:98) v h ) = ( Π V q − q , r h ) T h + (Π W u t − u t , v h ) T h , (6.3) for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . The next step is the consideration of the dual problem (3.1), which we again repeat for conve-nience: Let Φ + ∇ Ψ = 0 in Ω , ∇ · Φ = Θ in Ω , Ψ = 0 on ∂ Ω . (6.4)By the assumption at the beginning of Section 3, this boundary value problem admits the regularityestimate (cid:107) Φ (cid:107) H (Ω) + (cid:107) Ψ (cid:107) H (Ω) ≤ C (cid:107) Θ (cid:107) L (Ω) , (6.5)for all Θ ∈ L (Ω). Lemma 6.2.
We have (cid:107) ε uh (cid:107) T h ≤ Ch min { k, } ( (cid:107) q − Π V q (cid:107) T h + (cid:107) u t − Π W u t (cid:107) T h ) (cid:107) ε q h (cid:107) T h ≤ C ( (cid:107) q − Π V q (cid:107) T h + (cid:107) u t − Π W u t (cid:107) T h ) . Proof.
Let Θ = ε uh in the dual problem (6.4), and take ( r h , v h , (cid:98) v h ) = ( − Π V Φ , Π W Ψ , P M Ψ) in thedefinition of B (3.8) to get B ( ε q h , ε uh , ε (cid:98) uh ; − Π V Φ , Π W Ψ , P M Ψ)= − ( ε q h , Π V Φ ) T h + ( ε uh , ∇ · Π V Φ ) T h − (cid:104) ε (cid:98) uh , Π V Φ · n (cid:105) ∂ T h + ( ∇ · ε q h , Π W Ψ) T h − (cid:104) ε q h · n , P M Ψ (cid:105) ∂ T h + (cid:68) τ ( ε uh − ε (cid:98) uh ) , Π W Ψ − P M Ψ (cid:69) ∂ T h = − ( ε q h , Φ ) T h + ( ε q h , Φ − Π V Φ ) T h + ( ε uh , ∇ · Φ ) T h − ( ε uh , ∇ · ( Φ − Π V Φ )) T h + (cid:104) ε (cid:98) uh , ( Φ − Π V Φ ) · n (cid:105) ∂ T h + ( ∇ · ε q h , Ψ) T h + ( ∇ · ε q h , Π W Ψ − Ψ) T h − (cid:104) ε q h · n , Ψ (cid:105) ∂ T h + (cid:68) τ ( ε uh − ε (cid:98) uh ) , Π W Ψ − P M Ψ (cid:69) ∂ T h = ( ε q h , Φ − Π V Φ ) T h + (cid:107) ε uh (cid:107) T h . On the other hand, take ( r h , v h , (cid:98) v h ) = ( − Π V Φ , Π W Ψ , P M Ψ) in (6.3) to get B ( ε q h , ε uh , ε (cid:98) uh ; − Π V Φ , Π W Ψ , P M Ψ) = ( q − Π V q , Π V Φ ) T h + (Π W u t − u t , Π W Ψ) T h . (6.6)20uperconvergent interpolatory HDG methods for reaction diffusion IComparing the above two equalities gives (cid:107) ε uh (cid:107) T h = − ( ε q h , Φ − Π V Φ ) T h + ( q − Π V q , Π V Φ ) T h + (Π W u t − u t , Π W Ψ) T h = − ( ε q h , Φ − Π V Φ ) T h + ( q − Π V q , Π V Φ − Φ ) T h + ( q − Π V q , Φ ) T h + (Π W u t − u t , Π W Ψ) T h = − ( ε q h , Φ − Π V Φ ) T h + ( q − Π V q , Π V Φ − Φ ) T h − ( q − Π V q , ∇ Ψ) T h + (Π W u t − u t , Π W Ψ) T h = − ( ε q h , Φ − Π V Φ ) T h + ( q − Π V q , Π V Φ − Φ ) T h − ( q − Π V q , ∇ (Ψ − Π W Ψ)) T h + (Π W u t − u t , Π W Ψ − min { k, } Π Ψ) T h . Hence, by the regularity of the dual PDE (6.5), we have (cid:107) ε uh (cid:107) T h ≤ Ch (cid:107) ε q h (cid:107) T h + Ch min { k, } (cid:107) q − Π V q (cid:107) T h + Ch min { k, } (cid:107) u t − Π W u t (cid:107) T h . (6.7)Next, take ( r h , v h , (cid:98) v h ) = ( ε q h , ε uh , ε (cid:98) uh ) in (6.3) to obtain (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h = ( Π V q − q , ε q h ) T h + (Π W u t − u t , ε uh ) T h ≤ C (cid:107) Π V q − q (cid:107) T h + 12 (cid:107) ε q h (cid:107) T h + 4 C (cid:107) Π W u t − u t (cid:107) T h + 14 C (cid:107) ε uh (cid:107) T h . This implies (cid:107) ε q h (cid:107) T h + (cid:104) τ ( ε uh − ε (cid:98) uh ) , ε uh − ε (cid:98) uh (cid:105) ∂ T h ≤ C (cid:107) Π V q − q (cid:107) T h + 8 C (cid:107) Π W u t − u t (cid:107) T h + 12 C (cid:107) ε uh (cid:107) T h . (6.8)Next, use h ≤ Lemma 6.3.
We have (cid:107) ∂ t (Π W u − u h ) (cid:107) T h ≤ Ch min { k, } ( (cid:107) q t − Π V q t (cid:107) T h + (cid:107) u tt − Π W u tt (cid:107) T h ) . Acknowledgements
G. Chen thanks Missouri University of Science and Technology for hosting him as a visiting scholar;some of this work was completed during his research visit. J. Singler and Y. Zhang were supportedin part by National Science Foundation grant DMS-1217122. J. Singler and Y. Zhang thank theIMA for funding research visits, during which some of this work was completed.
References [1] Susanne C. Brenner and L. Ridgway Scott.
The mathematical theory of finite element methods ,volume 15 of
Texts in Applied Mathematics . Springer, New York, third edition, 2008.[2] Aycil Cesmelioglu, Bernardo Cockburn, and Weifeng Qiu. Analysis of a hybridizable discon-tinuous Galerkin method for the steady-state incompressible Navier-Stokes equations.
Math.Comp. , 86(306):1643–1670, 2017. 21. Chen, B. Cockburn, J. Singler, and Y. Zhang[3] Brandon Chabaud and Bernardo Cockburn. Uniform-in-time superconvergence of HDG meth-ods for the heat equation.
Math. Comp. , 81(277):107–129, 2012.[4] Chuan Miao Chen, Stig Larsson, and Nai Ying Zhang. Error estimates of optimal order forfinite element methods with interpolated coefficients for the nonlinear heat equation.
IMA J.Numer. Anal. , 9(4):507–524, 1989.[5] G. Chen, B. Cockburn, J. R. Singler, and Y. Zhang. Superconvergent Group HDG methodsfor reaction diffusion equations II: HHO-inspired methods. In preparation.[6] Zhangxin Chen and Jim Douglas, Jr. Approximation of coefficients in hybrid and mixedmethods for nonlinear parabolic problems.
Mat. Apl. Comput. , 10(2):137–160, 1991.[7] I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna. Product approximation fornonlinear problems in the finite element method.
IMA J. Numer. Anal. , 1(3):253–266, 1981.[8] B. Cockburn, J. R. Singler, and Y. Zhang. Interpolatory HDG Methods for Parabolic Semi-linear PDEs.
J. Sci. Comput . In revision.[9] Bernardo Cockburn, Daniele A. Di Pietro, and Alexandre Ern. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods.
ESAIM Math. Model. Numer. Anal. ,50(3):635–650, 2016.[10] Bernardo Cockburn and Guosheng Fu. Superconvergence by M -decompositions. Part II: Con-struction of two-dimensional finite elements. ESAIM Math. Model. Numer. Anal. , 51(1):165–186, 2017.[11] Bernardo Cockburn and Guosheng Fu. Superconvergence by M -decompositions. Part III: Con-struction of three-dimensional finite elements. ESAIM Math. Model. Numer. Anal. , 51(1):365–398, 2017.[12] Bernardo Cockburn, Guosheng Fu, and Francisco-Javier. Sayas. Superconvergence by M -decompositions. Part I: General theory for HDG methods for diffusion. Math. Comp. ,86(306):1609–1641, 2017.[13] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridizationof discontinuous Galerkin, mixed, and continuous Galerkin methods for second order ellipticproblems.
SIAM J. Numer. Anal. , 47(2):1319–1365, 2009.[14] Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, andFrancisco-Javier Sayas. Analysis of HDG methods for Stokes flow.
Math. Comp. , 80(274):723–760, 2011.[15] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas. A projection-based error analysis of HDG methods.
Math. Comp. , 79(271):1351–1367, 2010.[16] Bernardo Cockburn and Jiguang Shen. A hybridizable discontinuous Galerkin method for the p -Laplacian. SIAM J. Sci. Comput. , 38(1):A545–A566, 2016.[17] Benjamin T. Dickinson and John R. Singler. Nonlinear model reduction using group properorthogonal decomposition.
Int. J. Numer. Anal. Model. , 7(2):356–372, 2010.[18] Jim Douglas, Jr. and Todd Dupont. The effect of interpolating the coefficients in nonlinearparabolic Galerkin procedures.
Math. Comput. , 20(130):360–389, 1975.22uperconvergent interpolatory HDG methods for reaction diffusion I[19] C. A. J. Fletcher. The group finite element formulation.
Comput. Methods Appl. Mech. Engrg. ,37(2):225–244, 1983.[20] C. A. J. Fletcher. Time-splitting and the group finite element formulation. In
Computa-tional techniques and applications: CTAC-83 (Sydney, 1983) , pages 517–532. North-Holland,Amsterdam, 1984.[21] Hardik Kabaria, Adrian J. Lew, and B. Cockburn. A hybridizable discontinuous Galerkinformulation for non-linear elasticity.
Comput. Methods Appl. Mech. Engrg. , 283:303–329, 2015.[22] Dongho Kim, Eun-Jae Park, and Boyoon Seo. Two-scale product approximation for semilinearparabolic problems in mixed methods.
J. Korean Math. Soc. , 51(2):267–288, 2014.[23] Stig Larsson, Vidar Thom´ee, and Nai Ying Zhang. Interpolation of coefficients and transfor-mation of the dependent variable in finite element methods for the nonlinear heat equation.
Math. Methods Appl. Sci. , 11(1):105–124, 1989.[24] J. C. L´opez Marcos and J. M. Sanz-Serna. Stability and convergence in numerical analysis.III. Linear investigation of nonlinear stability.
IMA J. Numer. Anal. , 8(1):71–84, 1988.[25] D. Moro, N.-C. Nguyen, and J. Peraire. A hybridized discontinuous Petrov-Galerkin schemefor scalar conservation laws.
Internat. J. Numer. Methods Engrg. , 91:950–970, 2012.[26] N. C. Nguyen and J. Peraire. Hybridizable discontinuous Galerkin methods for partial differ-ential equations in continuum mechanics.
J. Comput. Phys. , 231:5955–5988, 2012.[27] N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuousGalerkin method for linear convection-diffusion equations.
J. Comput. Phys. , 228(9):3232–3254, 2009.[28] N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuousGalerkin method for nonlinear convection-diffusion equations.
J. Comput. Phys. , 228(23):8841–8855, 2009.[29] N. C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin methodfor the incompressible Navier-Stokes equations (AIAA Paper 2010-362). In
Proceedings of the48th AIAA Aerospace Sciences Meeting and Exhibit , Orlando, Florida, January 2010.[30] N. C. Nguyen, J. Peraire, and B. Cockburn. A class of embedded discontinuous Galerkinmethods for computational fluid dynamics.
J. Comput. Phys. , 302:674–692, 2015.[31] J. Peraire, N. C. Nguyen, and B. Cockburn. A hybridizable discontinuous Galerkin method forthe compressible Euler and Navier-Stokes equations (AIAA Paper 2010-363). In
Proceedingsof the 48th AIAA Aerospace Sciences Meeting and Exhibit , Orlando, Florida, January 2010.[32] J. M. Sanz-Serna and L. Abia. Interpolation of the coefficients in nonlinear elliptic Galerkinprocedures.
SIAM J. Numer. Anal. , 21(1):77–83, 1984.[33] Rolf Stenberg. Postprocessing schemes for some mixed finite elements.
RAIRO Mod´el. Math.Anal. Num´er. , 25(1):151–167, 1991.[34] Yves Tourigny. Product approximation for nonlinear Klein-Gordon equations.
IMA J. Numer.Anal. , 10(3):449–462, 1990. 23. Chen, B. Cockburn, J. Singler, and Y. Zhang[35] Cheng Wang. Convergence of the interpolated coefficient finite element method for the two-dimensional elliptic sine-Gordon equations.
Numer. Methods Partial Differential Equations ,27(2):387–398, 2011.[36] Zhu Wang. Nonlinear model reduction based on the finite element method with interpolatedcoefficients: semilinear parabolic equations.
Numer. Methods Partial Differential Equations ,31(6):1713–1741, 2015.[37] Mary Fanett Wheeler. A priori L error estimates for Galerkin approximations to parabolicpartial differential equations. SIAM J. Numer. Anal. , 10:723–759, 1973.[38] Ziqing Xie and Chuanmiao Chen. The interpolated coefficient FEM and its application incomputing the multiple solutions of semilinear elliptic problems.
Int. J. Numer. Anal. Model. ,2(1):97–106, 2005.[39] Zhiguang Xiong and Chuanmiao Chen. Superconvergence of rectangular finite element withinterpolated coefficients for semilinear elliptic problem.
Appl. Math. Comput. , 181(2):1577–1584, 2006.[40] Zhiguang Xiong and Chuanmiao Chen. Superconvergence of triangular quadratic finite ele-ment with interpolated coefficients for semilinear parabolic equation.
Appl. Math. Comput. ,184(2):901–907, 2007.[41] Zhiguang Xiong, Yanping Chen, and Yan Zhang. Convergence of FEM with interpolatedcoefficients for semilinear hyperbolic equation.
J. Comput. Appl. Math. , 214(1):313–317, 2008.[42] Jianfeng Zhu, Yong-Tao Zhang, Stuart A. Newman, and Mark Alber. Application of dis-continuous Galerkin methods for reaction-diffusion systems in developmental biology.