A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations
AA Superconvergent Ensemble HDG Method for ParameterizedConvection Diffusion Equations
Gang Chen ∗ Liangya Pi † Liwei Xu ‡ . Yangwen Zhang § March 12, 2019
Abstract
In this paper, we first devise an ensemble hybridizable discontinuous Galerkin (HDG) methodto efficiently simulate a group of parameterized convection diffusion PDEs. These PDEs havedifferent coefficients, initial conditions, source terms and boundary conditions. The ensembleHDG discrete system shares a common coefficient matrix with multiple right hand side (RHS)vectors; it reduces both computational cost and storage. We have two contributions in this paper.First, we derive an optimal L convergence rate for the ensemble solutions on a general polygonaldomain, which is the first such result in the literature. Second, we obtain a superconvergent ratefor the ensemble solutions after an element-by-element postprocessing under some assumptionson the domain and the coefficients of the PDEs. We present numerical experiments to confirmour theoretical results. A challenge in numerical simulations is to reduce computational cost while keeping accuracy. To-ward this end, many fast algorithms have been proposed, which include domain decompositionmethods [30], multigrid methods [38], interpolated coefficient methods [8, 16, 34], and so on. Thesemethods are only suitable for a single simulation, not for a group of simulations with differentcoefficients, initial conditions, source terms and boundary conditions in many scenarios; for exam-ple, one needs repeated simulations to obtain accurate statistical information about the outputs ofinterest in some uncertainty quantification problems. A common way is to treat the simulationsseperately; this requires computational effort and memory. Parallel computing is one method thatcan solve this problem if sufficient memory is available.However, the computational effort and storage requirement is still a great challenge in realsimulations. An ensemble method was proposed by Jiang and Layton [25] to address this issue.They studied a set of J solutions of the Navier-Stokes equations with different initial conditionsand forcing terms. This algorithm uses the mean of the solutions to form a common coefficientmatrix at each time step. Hence, the problem is reduced to solving one linear system with manyright hand side (RHS) vectors, which can be efficiently computed by many existing algorithms, ∗ School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu, China([email protected]). † Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA([email protected]). ‡ School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu, China([email protected]) § Department of Mathematics Science, University of Delaware, Newark, DE, USA ([email protected]). a r X i v : . [ m a t h . NA ] M a r . Chen, L. Pi, L. Xu Y. Zhangsuch as LU factorization, GMRES, etc. The ensemble scheme has been extended to many differentmodels; see, e.g., [17, 19–24, 26, 27]. Recently, Luo and Wang [28] extended this idea to a stochasticparabolic PDE. It is worthwhile to mention that all the above works only obtained suboptimal L convergence rate for the ensemble solutions.All the previous works have used continuous Galerkin (CG) methods; however, for high Reynoldsnumber flows [23,26,36] using a modified CG method may still cause non-physical oscillations. Theliterature on discontinuous Galerkin (DG) methods for simulating a single convection diffusionPDE is already substantial and the research in this area is still active; see, e.g. [1, 15, 37]. However,there are no theoretical or numerical analysis works on DG methods for the spatial discretizationof a group of parameterized convection diffusion equations.However, the number of degrees of freedom for DG methods is much larger compared to CGmethods; this is the main drawback of DG methods. Hybridizable discontinuous Galerkin (HDG)methods were originally proposed by Cockburn, Gopalakrishnan, and Lazarov in [9] to fix thisissue. The HDG methods are based on a mixed formulation and introduce a numerical flux and anumerical trace to approximate the flux and the trace of the solution. The discrete HDG globalsystem is only in terms of the numerical trace variable since we can element-by-element eliminatethe numerical flux and solution. Therefore, HDG methods have a significantly smaller numberof globally coupled degrees of freedom compareed to DG methods. Moreover, HDG methodskeep the advantages of DG methods, which are suitable for convection diffusion problems; see,e.g., [4–6, 18, 29]. Also, HDG methods have been applied to flow problems [2, 10, 12, 13, 13, 14, 31, 32]and hyperbolic equations [7, 33, 35].In this work, we propose a new Ensemble HDG method to investigate a group of parame-terized convection diffusion equations on a Lipschitz polyhedral domain Ω ⊂ R d ( d ≥ j = 1 , , · · · , J , find ( q j , u j ) satisfying c j q j + ∇ u j = 0 in Ω × (0 , T ] ,∂ t u j + ∇ · q j + β j · ∇ u j = f j in Ω × (0 , T ] ,u j = g j on ∂ Ω × (0 , T ] ,u j ( · ,
0) = u j in Ω , (1.1)where the vector vector fields β j satisfy ∇ · β j = 0 . (1.2)We make other smoothness assumptions on the data of system (1.1) for our analysis. The HDG Method.
To better describe the Ensemble HDG method, we first give the semidis-cretization of the system (1.1) use an existing HDG method [11]. Let T h be a collection of disjointsimplexes K that partition Ω and let ∂ T h be the set { ∂K : K ∈ T h } . Let e ∈ E oh be the interior faceif the Lebesgue measure of e = ∂K + ∩ ∂K − is non-zero, similarly, e ∈ E ∂h be the boundary face ifthe Lebesgue measure of e = ∂K ∩ ∂ Ω is non-zero. Finally, we set( 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 ( · , · ) K denotes the L ( K ) inner product and (cid:104)· , ·(cid:105) ∂K denotes the L inner product on ∂K .Let P k ( K ) denote the set of polynomials of degree at most k on the element K . We define the2 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equationsfollowing discontinuous finite element spaces V h := { v ∈ [ L (Ω)] d : v | K ∈ [ P k ( K )] d , ∀ K ∈ T h } ,W h := { w ∈ L (Ω) : w | K ∈ P k ( K ) , ∀ K ∈ T h } ,Z h := { z ∈ L (Ω) : z | K ∈ P k +1 ( K ) , ∀ K ∈ T h } ,M h := { µ ∈ L ( ε h ) : µ | e ∈ P k ( e ) , ∀ e ∈ E h , µ | E ∂h = 0 } . We use the notation ∇ v h and ∇ · r h to denote the gradient of v h ∈ W h and the divergence of r h ∈ V h applied piecewise on each element K ∈ T h .The semidiscrete HDG method finds ( q jh , u jh , (cid:98) u jh ) ∈ V h × W h × M h such that for all j =1 , · · · , J ( c j q jh , r h ) T h − ( u jh , ∇ · r h ) T h + (cid:104) (cid:98) u jh , r h · n (cid:105) ∂ T h = −(cid:104) g j , r h · n (cid:105) ε ∂h , ( ∂ t u jh , v h ) T h − ( q jh + β j u jh , ∇ v h ) T h + (cid:104) (cid:98) q jh · n , v h (cid:105) ∂ T h + (cid:104) β j · n (cid:98) u jh , v h (cid:105) ∂ T h + (cid:104) β j · n g j , v h (cid:105) ε ∂h = ( f j , v h ) T h , (cid:104) (cid:98) q jh · n + β j · n (cid:98) u jh , (cid:98) v h (cid:105) ∂ T h = 0 , (1.3)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . Here the numerical traces on ∂ T h are defined as (cid:98) q jh · n = q jh · n + τ j ( u jh − (cid:98) u jh ) on ∂ T h \ ε ∂h , (1.4) (cid:98) q jh · n = q jh · n + τ j ( u jh − g j ) on ε ∂h , (1.5)where τ j are positive stabilization functions defined on ∂ T h satisfying τ j = τ + β j · n on ∂ T h , and the function τ is a positive constant on each element K ∈ T h . The Ensemble HDG Method.
It is obvious to see that the system (1.3)-(1.5) has J differentcoefficient matrices. The idea of the Ensemble HDG method is to treat the system to share onecommon coefficient matrix by changing the variables c j and β j into their ensemble means. Beforewe define the Ensemble HDG method, we give some notation first.Suppose the time domain [0 , T ] is uniformly partition into N steps with time step ∆ t and let t n = n ∆ t for n = 1 , · · · , N . Moreover, ¯ c n and ¯ β n stand for the ensemble means of the inversecoefficient of diffusion and convection coefficient at time t n , respectively, defined by¯ c n = 1 J J (cid:88) j =1 c nj and ¯ β n = 1 J J (cid:88) j =1 β nj , (1.6)the superscript n denotes the function value at the time t n .Substitute (1.4)-(1.5) into (1.3), and use some simple algebraic manipulation, the ensemble mean(1.6), and the previous step to replace the current step to obtain the Ensemble HDG formulation:find ( q njh , u njh , (cid:98) u njh ) ∈ V h × W h × M h such that for all j = 1 , , · · · , J (¯ c n q njh , r h ) T h − ( u njh , ∇ · r h ) T h + (cid:104) (cid:98) u njh , r h · n (cid:105) ∂ T h = ((¯ c n − c nj ) q n − jh , r h ) T h −(cid:104) g nj , r h · n (cid:105) E ∂h , (1.7a)( ∂ + t u njh , v h ) T h + ( ∇ · q njh , v h ) T h − (cid:104) q njh · n , (cid:98) v h (cid:105) ∂ T h + ( β n · ∇ u njh , v h ) T h −(cid:104) β n · n , u njh (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( u njh − (cid:98) u njh ) , v h − (cid:98) v h (cid:105) ∂ T h = ( f nj , v h ) T h + (cid:104) τ g nj , v h (cid:105) E ∂h +(( β n − β nj ) · ∇ u n − jh , v h ) T h − (cid:104) ( β n − β nj ) · n , u n − jh (cid:98) v h (cid:105) ∂ T h , (1.7b)3. Chen, L. Pi, L. Xu Y. Zhangfor all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . The initial conditions u jh and q jh will be specified later.Finally, we let ∂ + t u njh = u njh − u n − jh ∆ t . It is easy to see that the system (1.7) shares one matrix with J RHS vectors, and it is moreefficient to solve than performing J separate simulations. It is worth mentioning that this is the first time that an ensemble scheme has been derived incorporating HDG methods; it is even the first time for DG methods. We provide a rigorous error analysis to obtain an optimal L convergencerate for the flux q j and the solution u j on general polygonal domain Ω in Section 3. To the best ofour knowledge, this is the first time in the literature. One of the excellent features of HDG methodsis that we can obtain superconvergence after an element-by-element postprocessing; we show thatthis result also holds in the Ensemble HDG algorithm under some conditions on the domain Ωand the velocity vector fields β j . This is also the first superconvergent ensemble algorithm in theliterature. Finally, some numerical experiments are presented to confirm our theoretical results inSection 4. Furthermore, we also present numerical results for convection dominated problems with c − j (cid:28) We begin with some notation. We use the standard notation W m,p ( D ) for Sobolev spaces on D with norm (cid:107) · (cid:107) m,p,D and seminorm | · | m,p,D . We also write H m ( D ) instead of W m, ( D ), and weomit the index p in the corresponding norms and seminorms. Also, we omit the index m when m = 0 in the corresponding norms and seminorms. Moreover, we drop the subscript D if there isno ambiguity in the statement. We denote by C (0 , T ; W m,s (Ω)) the Banach space of all continuousfunctions from [0 , T ] into W m,s (Ω), and L p (0 , T ; W m,s (Ω)) for 1 ≤ p ≤ ∞ is similarly defined.To obtain the stability of (1.7) in this section, we assume the data of (1.1) satisfies (A1): f j ∈ C (0 , T ; L (Ω)), g j ∈ C (0 , T ; H / ( ∂ Ω)), u j ∈ L (Ω), c j ∈ C (0 , T ; L ∞ (Ω)) and thevector fields β j ∈ C (0 , T ; W , ∞ (Ω)). (A2): There exists a postive constant c such that c nj ≥ c , and the ensemble mean satisfies thecondition | ¯ c n − c nj | < min { ¯ c n , ¯ c n − } , ∀ x ∈ Ω and 1 ≤ n ≤ N, ≤ j ≤ J. (2.1)It is worth mentioning that we don’t assume any conditions like (2.1) on the functions β j . Thefunction τ is a piecewise constant function independent of j satisfyingmin ≤ j ≤ J ( τ + 12 β j · n ) ≥
12 max ≤ j ≤ J (cid:107) β j (cid:107) , ∞ , ∀ x ∈ ∂ T h . (2.2)Next, let Π (cid:96) and P M denote the standard L projection operators Π (cid:96) : L ( K ) → P (cid:96) ( K ) and P M : L ( e ) → P k ( e ) satisfying(Π (cid:96) w, v h ) K = ( w, v h ) K , ∀ v h ∈ P (cid:96) ( K ) , (2.3a) (cid:104) P M w, (cid:98) v h (cid:105) e = (cid:104) w, (cid:98) v h (cid:105) e , ∀ (cid:98) v h ∈ P k ( e ) . (2.3b)The following error estimates for the L projections are standard:4 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations Lemma 1.
Suppose k, (cid:96) ≥
0. There exists a constant C independent of K ∈ T h such that (cid:107) w − Π (cid:96) w (cid:107) K ≤ Ch (cid:96) +1 | w | (cid:96) +1 ,K , ∀ w ∈ H (cid:96) +1 ( K ) , (2.4a) (cid:107) w − P M w (cid:107) ∂K ≤ Ch k +1 / | w | k +1 ,K ∀ w ∈ H k +1 ( K ) . (2.4b)Moreover, the vector L projection Π (cid:96) is defined similarly.We choose the initial conditions u jh = Π k +1 u , q jh = −∇ u jh /c j . To make the presentationsimple for the stability, we assume g j = 0 for j = 1 , · · · , J in this section. Lemma 2.
If condition (2.1) holds, then the Ensemble HDG formulation is unconditionally stableand we have the following estimate:max ≤ n ≤ N (cid:107) u njh (cid:107) T h + N (cid:88) n =1 (cid:107) u njh − u n − jh (cid:107) T h + ∆ t N (cid:88) n =1 (cid:107)√ ¯ c n q njh (cid:107) T h + (cid:107)√ τ ( u njh − (cid:98) u njh ) (cid:107) ∂ T h ≤ C ∆ t N (cid:88) n =1 (cid:107) f nj (cid:107) T h + C (cid:107) u jh (cid:107) T h + C (cid:107) q jh (cid:107) T h . Proof.
Take ( r h , v h , (cid:98) v h ) = ( q njh , u njh , (cid:98) u njh ) in (1.7), use the polarization identity( a − b ) a = 12 ( a − b + ( a − b ) ) , (2.5)and add the Equation (1.7a) and Equation (1.7b) together to give (cid:107) u njh (cid:107) T h − (cid:107) u n − jh (cid:107) T h t + (cid:107) u njh − u n − jh (cid:107) T h t + (cid:107)√ ¯ c n q njh (cid:107) T h + (cid:107)√ τ ( u njh − (cid:98) u njh ) (cid:107) ∂ T h = − ( β n · ∇ u njh , u njh ) T h + (cid:104) ( β n · n ) u njh , (cid:98) u njh (cid:105) ∂ T h + (( c n − c nj ) q n − jh , q njh ) T h + (( β n − β nj ) · ∇ u n − jh , u njh ) T h − (cid:104) ( β n − β nj ) · n , u n − jh (cid:98) u njh (cid:105) ∂ T h + ( f nj , u njh ) T h . (2.6)By Green’s formula and the fact (cid:104) ( β n · n ) (cid:98) u njh , (cid:98) u njh (cid:105) ∂ T h = 0, we have − ( β n · ∇ u njh , u njh ) T h + (cid:104) ( β n · n ) u njh , (cid:98) u njh (cid:105) ∂ T h ≤ (cid:107) (cid:113) | β n · n | ( u njh − (cid:98) u njh ) (cid:107) ∂ T h . Hence, condition (2.2) gives (cid:107) u njh (cid:107) T h − (cid:107) u n − jh (cid:107) T h t + (cid:107) u njh − u n − jh (cid:107) T h t + (cid:107)√ ¯ c n q njh (cid:107) T h + 12 (cid:107)√ τ ( u njh − (cid:98) u njh ) (cid:107) ∂ T h ≤ (( c n − c nj ) q n − jh , q njh ) T h + (( β n − β nj ) · ∇ u n − jh , u njh ) T h − (cid:104) ( β n − β nj ) · n , u n − jh (cid:98) u nh (cid:105) ∂ T h + ( f nj , u njh ) T h = R + R + R + R . Next, we estimate { R i } i =1 . First, by the condition (2.1), there exist 0 < α < R = (( c n − c nj ) q n − jh , q njh ) T h ≤ α (cid:107)√ ¯ c n q njh (cid:107) T h + α (cid:107)√ ¯ c n − q n − jh (cid:107) T h .
5. Chen, L. Pi, L. Xu Y. ZhangThe term R + R needs a detailed argument. For simplicity, let γ = β n − β nj . We have R + R = ( γ · ∇ u n − jh , u njh ) T h − (cid:104) γ · n , u n − jh (cid:98) u njh (cid:105) ∂ T h = (( γ − Π γ ) · ∇ u n − jh , u njh ) T h − (cid:104) ( γ − Π γ ) · n , u n − jh (cid:98) u njh (cid:105) ∂ T h + ( Π γ · ∇ u n − jh , u njh ) T h − (cid:104) Π γ · n , u n − jh (cid:98) u njh (cid:105) ∂ T h = (( γ − Π γ ) · ∇ u n − jh , u njh ) T h − (cid:104) ( γ − Π γ ) · n , u n − jh (cid:98) u njh (cid:105) ∂ T h + (¯ c n q njh , Π γ u n − jh ) T h − ((¯ c n − c nj ) q n − jh , Π γ u n − jh ) T h , where we used Equation (1.7a) in the last identity. Hence, R + R = (( γ − Π γ ) · ∇ u n − jh , u njh ) T h − (cid:104) ( γ − Π γ ) · n , u n − jh (cid:98) u njh (cid:105) ∂ T h + (¯ c n q njh , Π γ u n − jh ) T h − ((¯ c n − c nj ) q n − jh , Π γ u n − jh ) T h ≤ (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,K (cid:107)∇ u n − jh (cid:107) K (cid:107) u njh (cid:107) K + (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,∂K (cid:107) u n − jh (cid:107) ∂K ( (cid:107) (cid:98) u njh − u njh (cid:107) ∂K + (cid:107) u njh (cid:107) ∂K )+ (cid:107) Π γ (cid:107) ∞ , T h (cid:107) ¯ c n q njh (cid:107) T h (cid:107) u n − jh (cid:107) T h + (cid:107) (¯ c n − c nj ) Π γ (cid:107) ∞ , T h (cid:107) q n − jh (cid:107) T h (cid:107) u n − jh (cid:107) T h = R + R + R + R . For R , use the local inverse inequality: R ≤ C (cid:88) K ∈T h h K (cid:107) γ (cid:107) , ∞ ,K h − K (cid:107) u n − jh (cid:107) K (cid:107) u njh (cid:107) K ≤ C ( (cid:107) u n − jh (cid:107) T h + (cid:107) u njh (cid:107) T h ) . Apply the trace inequality and inverse inequality for the term R to give R ≤ C (cid:88) K ∈T h h K (cid:107) γ (cid:107) , ∞ ,K h − / K (cid:107) u n − jh (cid:107) K ( (cid:107) (cid:98) u njh − u njh (cid:107) ∂K + h − / K (cid:107) u njh (cid:107) K ) ≤ C ( (cid:107) u n − jh (cid:107) T h + (cid:107) u njh (cid:107) T h ) + 14 (cid:107)√ τ ( (cid:98) u njh − u njh ) (cid:107) ∂ T h . For the terms R and R , use Young’s inequality to obtain R ≤ − α (cid:107)√ ¯ c n q njh (cid:107) T h + C (cid:107) u n − jh (cid:107) T h ,R ≤ − α (cid:107)√ ¯ c n − q n − jh (cid:107) T h + C (cid:107) u n − jh (cid:107) T h . The Cauchy-Schwarz inequality for the term R gives R = ( f nj , u njh ) T h ≤
12 ( (cid:107) f nj (cid:107) T h + (cid:107) u njh (cid:107) T h ) . We add (2.6) from n = 1 to n = N , and use the above inequalities to getmax ≤ n ≤ N (cid:107) u njh (cid:107) T h + N (cid:88) n =1 (cid:107) u njh − u n − jh (cid:107) T h + ∆ t N (cid:88) n =1 (cid:107)√ ¯ c n q njh (cid:107) T h + (cid:107)√ τ ( u njh − (cid:98) u njh ) (cid:107) ∂ T h ≤ C ∆ t N (cid:88) n =1 (cid:107) e u n jh (cid:107) T h + C ∆ t N (cid:88) n =1 (cid:107) f nj (cid:107) T h + C (cid:107) u jh (cid:107) T h + C (cid:107) q jh (cid:107) T h . Gronwall’s inequality applied to the above inequality gives the desired result.6 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations
The strategy of the error analysis for the Ensemble HDG method is based on [3] and [5]. First, wedefine the HDG projections, and use an energy argument to obtain an optimal convergence rate forthe ensemble solutions. Second, we define an HDG elliptic projection as in [3], which is a crucialstep to get the superconvergence. Next, we give our main results, and in the end, we provide arigorous error estimation for our Ensemble HDG method.Throughout, we assume the data and the solution of (1.1) are smooth enough, and the initialconditions ( q jh , u jh ) of the Ensemble HDG system (1.7) are chosen as in Section 2. For any t ∈ [0 , T ], let ( Π jV q j , Π jW u j ) be the HDG projection of ( q j , u j ), where Π jV q j and Π jW u j denote components of the HDG projection of q j and u j into V h and W h , respectively. On eachelement K ∈ T h , ( Π jV q j , Π jW u j ) satisfy the following equations( Π jV q j + β j Π jW u j , r ) K = ( q j + β j u j , r ) K , (3.1a)(Π jW u j , w ) K = ( u j , w ) K , (3.1b) (cid:104) Π jV q j · n + β j · n Π jW u j + τ Π jW u j , µ (cid:105) e = (cid:104) q j · n + β j · n u j + τ u j , µ (cid:105) e , (3.1c)for all ( r , w, µ ) ∈ [ P k − ( K )] d × P k − ( K ) × P k ( e ) and for all faces e of the simplex K . We noticethe projections are only determined by (3.1c) when k = 0. The proof of the following lemma issimilar to a result established in [5] and hence is omitted. Lemma 3.
Suppose the polynomial degree satisfies k ≥ τ >
0. Then the system (3.1)is uniquely solvable for Π jV q j and Π jW u j . Furthermore, there is a constant C independent of K and τ such that for (cid:96) q j , (cid:96) u j in [0 , k ] (cid:107) Π jV q j − q j (cid:107) K ≤ Ch (cid:96) q j +1 K | q j | H (cid:96) q j +1 ( K ) + Ch (cid:96) uj +1 K | u j | H (cid:96)uj +1 ( K ) , (cid:107) Π jW u j − u j (cid:107) K ≤ Ch (cid:96) uj +1 K | u j | H (cid:96)uj +1 ( K ) + Ch (cid:96) q j +1 K |∇ · q j | H (cid:96) q j ( K ) . We can now state our main result for the Ensemble HDG method.
Theorem 1.
Let ( q nj , u nj ) and ( q njh , u njh ) be the solution of (1.1) at time t n and (1.7), respectively.If the coefficients c j satisfy (2.1), then we havemax ≤ n ≤ N (cid:107) u nj − u njh (cid:107) T h ≤ C ( h k +1 + ∆ t ) , (3.3a) (cid:118)(cid:117)(cid:117)(cid:116) ∆ t N (cid:88) n =1 (cid:107) q nj − q njh (cid:107) T h ≤ C ( h k +1 + ∆ t ) . (3.3b)Moreover, if k ≥
1, the elliptic regularity inequality (6.4) holds and the coefficients of the PDEsare independent of time, then we have (cid:118)(cid:117)(cid:117)(cid:116) ∆ t N (cid:88) n =1 (cid:107) u nj − u n(cid:63)jh (cid:107) T h ≤ C ( h k +2 + ∆ t ) , (3.4)where u n(cid:63)jh is the postprocessed approximation defined in (3.17).7. Chen, L. Pi, L. Xu Y. Zhang Remark 1.
To the best of our knowledge, all previous works only contain suboptimal L conver-gence rate for the ensemble solutions u j ; our result (3.3) is the first time to obtain the optimal L ∞ (0 , T ; L (Ω)) convergence rate on a general polygonal domain Ω. Moreover, if the coefficients ofthe PDEs are independent of time, then after an element-by-element postprocessing, we obtain thesuperconvergent rate (3.4) under some conditions on the domain; for example, a convex domain issufficient. This is also the first such result in the literature. (3.3) in Theorem 1 Lemma 4.
For all n = 1 , , · · · , N , we have the following equalities:( c nj Π jV q nj , r h ) T h − (Π jW u nj , ∇ · r h ) T h + (cid:104) P M u nj , r h · n (cid:105) ∂ T h = ( c nj ( Π jV q nj − q nj ) , r h ) T h , and ( ∇ · Π jV q nj , v h ) T h − (cid:104) Π jV q nj · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u nj − P M u nj ) , v h − (cid:98) v h (cid:105) ∂ T h + ( β nj · ∇ Π jW u nj , v h ) T h − (cid:104) β nj · n , (Π jW u nj ) (cid:98) v h (cid:105) ∂ T h = ( f nj − ∂ t u nj , v h ) T h , for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h and j = 1 , , · · · , J . Proof.
By the definitions of Π jW in (3.1b), P M in (2.3b), and the first equation (1.1), we get( c nj Π jV q nj , r h ) T h − (Π jW u nj , ∇ · r h ) T h + (cid:104) P M u nj , r h · n (cid:105) ∂ T h = ( c nj q nj , r h ) T h − (Π jW u nj , ∇ · r h ) T h + (cid:104) P M u nj , r h · n (cid:105) ∂ T h + ( c nj ( Π jV q nj − q nj ) , r h ) T h = ( c nj q nj , r h ) T h − ( u nj , ∇ · r h ) T h + (cid:104) u nj , r h · n (cid:105) ∂ T h + ( c nj ( Π jV q nj − q nj ) , r h ) T h = ( c nj q nj + ∇ u nj , r h ) T h + ( c nj ( Π jV q nj − q nj ) , r h ) T h = ( c nj ( Π jV q nj − q nj ) , r h ) T h . This proves the first identity.Next, we prove the second identity. First( ∇ · Π jV q nj , v h ) T h − (cid:104) Π jV q nj · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u nj − P M u nj ) , v h − (cid:98) v h (cid:105) ∂ T h + ( β j · ∇ Π jW u j , v h ) T h − (cid:104) β nj · n , (Π jW u nj ) (cid:98) v h (cid:105) ∂ T h = ( ∇ · q nj , v h ) T h + ( ∇ · ( Π jV q nj − q nj ) , v h ) T h − (cid:104) Π jV q nj · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u nj − P M u nj ) , v h − (cid:98) v h (cid:105) ∂ T h + ( β nj · ∇ u nj , v h ) T h + ( β nj · ∇ (Π jW u nj − u nj ) , v h ) T h − (cid:104) β nj · n , (Π jW u nj ) (cid:98) v h (cid:105) ∂ T h . By the definition of Π jV and Π jW in (3.1a) and ∇ · β nj = 0, we have( ∇ · ( Π jV q nj − q nj ) , v h ) T h + ( β nj · ∇ (Π jW u nj − u nj ) , v h ) T h = − ( Π jV q nj − q nj , ∇ v h ) T h + (cid:104) ( Π jV q nj − q nj ) · n , v h (cid:105) ∂ T h − ( β nj (Π jW u nj − u nj ) , ∇ v h ) T h + (cid:104) ( β nj · n )(Π jW u nj − u nj ) , v h (cid:105) ∂ T h = (cid:104) ( Π jV q nj − q nj ) · n , v h (cid:105) ∂ T h + (cid:104) ( β nj · n )(Π jW u nj − u nj ) , v h (cid:105) ∂ T h . ∇ · q nj , v h ) T h + ( β nj · ∇ u nj , v h ) T h = ( f nj − ∂ t u nj , v h ) T h and (3.1c), we have( ∇ · Π jV q nj , v h ) T h − (cid:104) Π jV q nj · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u nj − P M u nj ) , v h − (cid:98) v h (cid:105) ∂ T h + ( β j · ∇ Π jW u j , v h ) T h − (cid:104) β nj · n , (Π jW u nj ) (cid:98) v h (cid:105) ∂ T h = ( f nj − ∂ t u nj , v h ) T h + (cid:104) ( Π jV q nj − q nj ) · n , v h − (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u nj − P M u nj ) , v h − (cid:98) v h (cid:105) ∂ T h + (cid:104) ( β nj · n )(Π jW u nj − u nj ) , v h − (cid:98) v h (cid:105) ∂ T h = ( f nj − ∂ t u nj , v h ) T h . Then, substracting the result of Lemma 4 from the Ensemble HDG system (1.7) gives thefollowing error equations.
Lemma 5.
For η u n jh = u njh − Π jW u nj , η q n jh = q njh − Π jV q nj and η (cid:98) u n jh = (cid:98) u njh − P M u nj , for all j = 1 , , · · · , J ,we have the following error equations:(¯ c n η q n jh , r h ) T h − ( η u n jh , ∇ · r h ) T h + (cid:104) η (cid:98) u n jh , r h · n (cid:105) ∂ T h = ((¯ c n − c nj )( q n − jh − Π jV q nj ) , r h ) T h − ( c nj ( Π jV q nj − q nj ) , r h ) T h , (3.6a)and ( ∂ + t η u n jh , v h ) T h + ( ∇ · η q n jh , v h ) T h − (cid:104) η q n jh · n , (cid:98) v h (cid:105) ∂ T h + ( β n · ∇ η u n jh , v h ) T h − (cid:104) β n · n , η u n jh (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( η u n jh − η (cid:98) ujh ) , v h − (cid:98) v h (cid:105) ∂ T h = ( ∂ t u nj − ∂ + t Π jW u nj , v h ) T h + (( β n − β nj ) · ∇ ( u n − jh − Π jW u nj ) , v h ) T h − (cid:104) ( β nj − β nj ) · n , ( u n − jh − Π jW u nj ) (cid:98) v h (cid:105) ∂ T h , (3.6b)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h and n = 1 , , · · · , N . Lemma 6.
If condition (2.1) holds, then we have the following error estimate:max ≤ n ≤ N (cid:107) η u n jh (cid:107) T h + (cid:118)(cid:117)(cid:117)(cid:116) ∆ t N (cid:88) n =1 (cid:107)√ c n η q n jh (cid:107) T h ≤ C (cid:16) h k +1 + ∆ t (cid:17) . (3.7) Proof.
We take ( r h , v h , (cid:98) v h ) = ( η q n jh , η u n jh , η (cid:98) u n jh ) in (3.6), use the identity (2.5) and add Equation (3.6a)and Equation (3.15) together to get (cid:107) η u n jh (cid:107) T h − (cid:107) η u n − jh (cid:107) T h t + (cid:107) η u n jh − η u n − jh (cid:107) T h t + (cid:107)√ ¯ c n η q n jh (cid:107) T h + (cid:107)√ τ ( η u n jh − η (cid:98) u n jh ) (cid:107) ∂ T h = − ( β n · ∇ η u n jh , η u n jh ) T h + (cid:104) β n · n , η u n jh η (cid:98) u n jh (cid:105) ∂ T h + (( c n − c nj )( q n − jh − Π jV q nj ) , η q n jh ) T h + ( ∂ t u nj − ∂ + t Π jW u nj , η u n jh ) T h + (( β n − β nj ) · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) ( β n − β nj ) · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h . (3.8)By Green’s formula and the fact (cid:104) ( β n · n ) η (cid:98) u n jh , η (cid:98) u n jh (cid:105) ∂ T h = 0, we have( β n · ∇ η u n jh , η u n jh ) T h − (cid:104) β n · n , η u n jh η (cid:98) u n jh (cid:105) ∂ T h ≤ (cid:107) (cid:113) | β n · n | ( η u n jh − η (cid:98) u n jh ) (cid:107) ∂ T h .
9. Chen, L. Pi, L. Xu Y. ZhangCondition (2.2) and equality (3.8) give (cid:107) η u n jh (cid:107) T h − (cid:107) η u n − jh (cid:107) T h t + (cid:107) η u n jh − η u n − jh (cid:107) T h t + (cid:107)√ ¯ c n η q n jh (cid:107) T h + 12 (cid:107)√ τ ( η u n jh − η (cid:98) u n jh ) (cid:107) ∂ T h ≤ (( c n − c nj )( q n − jh − Π jV q nj ) , η q n jh ) T h + ( ∂ t u nj − ∂ + t Π jW u nj , η u n jh ) T h + (cid:104) (( β n − β nj ) · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h −(cid:104) ( β n − β nj ) · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h (cid:105) − ( c nj ( Π jV q nj − q nj ) , η q n jh ) T h = R + R + R + R . (3.9)Next, we estimate { R i } i =1 . By the condition (2.1), there exist 0 < α < R = (( c n − c nj )( η q n − jh − ∆ t∂ + t Π jV q nj ) , η q n jh ) T h ≤ α (cid:16) (cid:107)√ ¯ c n η q n jh (cid:107) T h + (cid:107)√ ¯ c n − η q n − jh (cid:107) T h (cid:17) + C ∆ t (cid:107) ∂ + t Π jV q nj (cid:107) T h ,R = ( ∂ + t ( u nj − Π jW u nj ) − ∂ + t u nj + ∂ t u nj , η u n jh ) T h ≤ C (cid:16) (cid:107) ∂ + t ( u nj − Π jW u nj ) (cid:107) T h + (cid:107) ∂ + t u nj − ∂ t u nj (cid:107) T h + (cid:107) η u n jh (cid:107) T h (cid:17) ,R ≤ − α (cid:107)√ ¯ c n η q n jh (cid:107) T h + Ch k +2 ( | u nj | k +1 + | q nj | k +1 ) . If we directly estimate R , we will obtain only suboptimal convergence rates. Therefore, we needa refined analysis for this term. For simplicity, let γ = β n − β nj . The following argument is similarto the proof of the stability Section 2; to make the proof self-contained, we include these detailshere. First R = ( γ · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) γ · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h = (( γ − Π γ ) · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) ( γ − Π γ ) · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h + ( Π γ · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) Π γ · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h . By the error equation (3.6a), we have( Π γ · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) Π γ · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h = ( ∇ · [ Π γ ( u n − jh − Π jW u nj )] , η u n jh ) T h − (cid:104) [( Π γ · n )( u n − jh − Π jW u nj )] , η (cid:98) u n jh (cid:105) ∂ T h = (¯ c n η q n jh , [ Π γ ( u n − jh − Π jW u nj )]) T h + ( c nj ( Π jV q nj − q nj ) , [ Π γ ( u n − jh − Π jW u nj )]) T h − ((¯ c n − c nj )( q n − jh − Π jV q nj ) , [ Π γ ( u n − jh − Π jW u nj )]) T h . This gives R = (( γ − Π γ ) · ∇ ( u n − jh − Π jW u nj ) , η u n jh ) T h − (cid:104) ( γ − Π γ ) · n , ( u n − jh − Π jW u nj ) η (cid:98) u n jh (cid:105) ∂ T h + (¯ c n η q n jh , Π γ ( u n − jh − Π jW u nj )) T h + ( c nj ( Π jV q nj − q nj ) , [ Π γ ( u n − jh − Π jW u nj )]) T h − ((¯ c n − c nj )( q n − jh − Π jV q nj ) , Π γ ( u n − jh − Π jW u nj )) T h .
10 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion EquationsHence, R ≤ (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,K (cid:107)∇ ( u n − jh − Π jW u nj ) (cid:107) K (cid:107) η u n jh (cid:107) K + (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,∂K (cid:107) u n − jh − Π jW u nj (cid:107) ∂K ( (cid:107) η (cid:98) u n jh − η u n jh (cid:107) ∂K + (cid:107) η u n jh (cid:107) ∂K )+ (cid:107) Π γ (cid:107) ∞ , T h (cid:107) ¯ c n η q n jh (cid:107) T h (cid:107) u n − jh − Π jW u nj (cid:107) T h + (cid:107) (¯ c n − c nj ) Π γ (cid:107) ∞ , T h (cid:107) q n − jh − Π jV q nj (cid:107) T h (cid:107) u n − jh − Π jW u nj (cid:107) T h + (cid:107) c nj Π γ (cid:107) ∞ , T h (cid:107) Π jV q nj − q nj (cid:107) T h (cid:107) u n − jh − Π jW u nj (cid:107) T h = R + R + R + R + R . For R , use the local inverse inequality: R ≤ C (cid:88) K ∈T h h K (cid:107) γ (cid:107) , ∞ ,K h − K (cid:107) u n − jh − Π jW u nj (cid:107) K (cid:107) η u n jh (cid:107) K ≤ C (cid:88) K ∈T h (cid:107) u n − jh − Π jW u nj (cid:107) K (cid:107) η u n jh (cid:107) K ≤ C ( (cid:107) η u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h + (cid:107) η u n jh (cid:107) T h ) . Apply the trace inequality and inverse inequality for the term R to give R ≤ C (cid:88) K ∈T h h K (cid:107) γ (cid:107) , ∞ ,K h − / K (cid:107) u n − jh − Π jW u nj (cid:107) K ( (cid:107) η (cid:98) u n jh − η u n jh (cid:107) ∂K + h − / K (cid:107) η u n jh (cid:107) K ) ≤ C (cid:88) K ∈T h (cid:107) u n − jh − Π jW u nj (cid:107) K ( (cid:107) η (cid:98) u n jh − η u n jh (cid:107) ∂K + (cid:107) η u n jh (cid:107) K ) ≤ C ( (cid:107) η u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h + (cid:107) η u n jh (cid:107) T h ) + 14 (cid:107)√ τ ( η (cid:98) u n jh − η u n jh ) (cid:107) ∂ T h . For the terms R , R , R and R , use Young’s inequality to obtain R ≤ − α (cid:107)√ ¯ c n η q n jh (cid:107) T h + C ( (cid:107) η u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h ) ,R ≤ − α (cid:107)√ ¯ c n η q n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jV q nj (cid:107) T h + C ( (cid:107) η u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h ) ,R ≤ Ch k +2 ( | u nj | k +1 + | q nj | k +1 ) + C ( (cid:107) η u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h ) . We add (3.9) from n = 1 to n = N , use the above inequalities to getmax ≤ n ≤ N (cid:107) η u n jh (cid:107) T h + ∆ t N (cid:88) n =1 (cid:107)√ ¯ c n η q n jh (cid:107) T h ≤ C ∆ t N (cid:88) n =1 (cid:107) η u n jh (cid:107) T h + C N (cid:88) n =1 (∆ t (cid:107) ∂ + t Π jW u nj (cid:107) T h + ∆ t (cid:107) ∂ + t Π jV q nj (cid:107) T h )+ C N (cid:88) n =1 (∆ t (cid:107) ∂ + t ( u nj − Π jW u nj ) (cid:107) T h + ∆ t (cid:107) ∂ + t u nj − ∂ t u nj (cid:107) T h )+ Ch k +2 N (cid:88) n =1 ∆ t ( | u nj | k +1 + | q nj | k +1 ) + (cid:107) η u jh (cid:107) T h + (cid:107) η q jh (cid:107) T h . (3.10)11. Chen, L. Pi, L. Xu Y. ZhangNow we move to bound the terms on the right side of the above inequality as follows,∆ t N (cid:88) n =1 (cid:107) ∂ + t Π jW u nj (cid:107) T h = ∆ t N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ∂ t Π jW u nj dt (cid:21) ≤ C ∆ t (cid:107) ∂ t Π jW u nj (cid:107) L (0 ,T ; L (Ω)) , ∆ t N (cid:88) n =1 (cid:107) ∂ + t Π jV q nj (cid:107) T h = ∆ t N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ∂ t Π jV q nj dt (cid:21) ≤ C ∆ t (cid:107) ∂ t Π jV q nj (cid:107) L (0 ,T ; L (Ω)) , and ∆ t N (cid:88) n =1 (cid:107) ∂ + t ( u nj − Π jW u nj ) (cid:107) T h = ∆ t − N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ∂ t ( u nj − Π jW u j dt ) (cid:21) ≤ C (cid:107) ∂ t ( u nj − Π jW u j ) (cid:107) L (0 ,T ; L (Ω)) , ∆ t N (cid:88) n =1 (cid:107) ∂ + t u nj − ∂ t u nj (cid:107) T h = ∆ t − N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ( t − t n − ) ∂ tt u j dt (cid:21) ≤ C ∆ t (cid:107) ∂ tt u j (cid:107) L (0 ,T ; L (Ω)) . Gronwall’s inequality and the estimates above applied to (3.10) give the result.From Lemma 6 and the estimate in Lemma 3 we complete the proof of (3.3) in Theorem 1. (3.4) in Theorem 1
To prove (3.4) in Theorem 1, we follow a similar strategy taken by Chen, Cockburn, Singler andZhang [3] and introduce an HDG elliptic projection in Section 3.4.1. We first bound the errorbetween the solutions of the HDG elliptic projection and the exact solution of the system (1.1).Then we bound the error between the solutions of the HDG elliptic projection and the EnsembleHDG problem (1.7). A simple application of the triangle inequality then gives a bound on theerror between the solutions of the Ensemble HDG problem and the system (1.1). We note thatthe coefficients of the PDEs are independent of time throughout this section. Hence, we drop thesuperscript n from c nj , β nj and the ensemble means c n , β n . For any t ∈ [0 , T ], let ( q jh , u jh , (cid:98) u jh ) ∈ V h × W h × M h be the solutions of the following steady stateproblems ( c j q jh , r h ) T h − ( u jh , ∇ · r h ) T h + (cid:104) (cid:98) u jh , r h · n (cid:105) ∂ T h = − (cid:104) g j , r h · n (cid:105) E ∂h , (3.11a)( ∇ · q jh , v h ) T h − (cid:104) q jh · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( u jh − (cid:98) u jh ) , v h − (cid:98) v h (cid:105) ∂ T h +( β j · ∇ u jh , v h ) T h − (cid:104) β j · n , u jh (cid:98) v h (cid:105) ∂ T h = ( f j − Π jW ∂ t u j , v h ) T h + (cid:104) τ g j , v h (cid:105) E ∂h , (3.11b)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h and j = 1 , , · · · , J .The proofs of the following estimates are given in Section 6.12 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations Theorem 2.
For any t ∈ [0 , T ] and for all j = 1 , , · · · , J , we have (cid:107) Π jV q j − q jh (cid:107) T h ≤ C A j , (3.12a) (cid:107) Π jW u j − u jh (cid:107) T h ≤ Ch min { k, } A j , (3.12b) (cid:107) ∂ t ( Π jV q j − q jh ) (cid:107) T h ≤ C B j , (3.12c) (cid:107) ∂ t (Π jW u j − u jh ) (cid:107) T h ≤ Ch min { k, } B j , (3.12d) (cid:107) ∂ tt (Π jW u j − u jh ) (cid:107) T h ≤ Ch min { k, } C j , (3.12e)where A j = (cid:107) u j − Π jW u j (cid:107) T h + (cid:107) q j − Π jV q j (cid:107) T h + (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h , B j = (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h + (cid:107) ∂ t q j − Π jV ∂ t q j (cid:107) T h + (cid:107) ∂ tt u j − Π jW ∂ tt u j (cid:107) T h , C j = (cid:107) ∂ tt u j − Π jW ∂ tt u j (cid:107) T h + (cid:107) ∂ tt q j − Π jV ∂ tt q j (cid:107) T h + (cid:107) ∂ ttt u j − Π jW ∂ ttt u j (cid:107) T h . Note that Theorem 2 bounds the error between the HDG elliptic projection of the solutions andthe exact solutions of the system (1.1). In the next three steps, we are going to bound the errorbetween the HDG elliptic projection of the ensemble solutions and the solutions of the EnsembleHDG problem (1.7).
For e u n jh = u njh − u njh , e q n jh = q njh − q njh and e (cid:98) u n jh = (cid:98) u njh − (cid:98) u njh , for all j = 1 , , · · · , J , wehave the following error equations(¯ ce q n jh , r h ) T h − ( e u n jh , ∇ · r h ) T h + (cid:104) e (cid:98) u n jh , r h · n (cid:105) ∂ T h = ((¯ c − c j )( q n − jh − q njh ) , r h ) T h , (3.13a)and ( ∂ + t e u n jh , v h ) T h + ( ∇ · e q n jh , v h ) T h − (cid:104) e q n jh · n , (cid:98) v h (cid:105) ∂ T h + ( β · ∇ e u n jh , v h ) T h − (cid:104) β · n , e u n jh (cid:98) v h (cid:105) ∂ T h + (cid:104) τ ( e u n jh − e (cid:98) ujh ) , v h − (cid:98) v h (cid:105) ∂ T h − ( ∂ + t u njh − ∂ t Π jW u nj , v h ) T h = (( β − β j ) · ∇ ( u n − jh − u njh ) , v h ) T h − (cid:104) ( β j − β j ) · n , ( u n − jh − u njh ) (cid:98) v h (cid:105) ∂ T h , (3.13b)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h and n = 1 , , · · · , N .The proof of Lemma 7 follows immediately by simply subtracting Equation (3.11) from Equa-tion (1.7). If condition (2.1) and the elliptic regularity inequality (6.4) holds, then we have thefollowing error estimate: max ≤ n ≤ N (cid:107) e u n jh (cid:107) T h ≤ C (cid:16) h k +1+min { k, } + ∆ t (cid:17) . (3.14) Proof.
The following proof is similar to the proof in Section 3.3; to make the proof self-contained,we include the details here. We take ( r h , v h , (cid:98) v h ) = ( e q n jh , e u n jh , e (cid:98) u n jh ) in (3.13), use the identity (2.5)13. Chen, L. Pi, L. Xu Y. Zhangand add Equation (3.13a) - Equation (3.13b) together to get (cid:107) e u n jh (cid:107) T h − (cid:107) e u n − jh (cid:107) T h t + (cid:107) e u n jh − e u n − jh (cid:107) T h t + (cid:107)√ ¯ ce q n jh (cid:107) T h + (cid:107)√ τ ( e u n jh − e (cid:98) u n jh ) (cid:107) ∂ T h = − ( β · ∇ e u n jh , e u n jh ) T h + (cid:104) β · n , e u n jh e (cid:98) u n jh (cid:105) ∂ T h + (( c − c j )( q n − jh − q njh ) , e q n jh ) T h + ( ∂ + t u njh − ∂ t Π jW u nj , e u n jh ) T h + (( β − β j ) · ∇ ( u n − jh − u njh ) , e u n jh ) T h − (cid:104) ( β − β j ) · n , ( u n − jh − u njh ) e (cid:98) u n jh (cid:105) ∂ T h . (3.15)By Green’s formula and the fact (cid:104) ( β · n ) e (cid:98) u n jh , e (cid:98) u n jh (cid:105) ∂ T h = 0, we have( β · ∇ e u n jh , e u n jh ) T h − (cid:104) β · n , e u n jh e (cid:98) u n jh (cid:105) ∂ T h ≤ (cid:107) (cid:113) | β · n | ( e u n jh − e (cid:98) u n jh ) (cid:107) ∂ T h . Condition (2.2) and equality (3.15) give (cid:107) e u n jh (cid:107) T h − (cid:107) e u n − jh (cid:107) T h t + (cid:107) e u n jh − e u n − jh (cid:107) T h t + (cid:107)√ ¯ ce q n jh (cid:107) T h + 12 (cid:107)√ τ ( e u n jh − e (cid:98) u n jh ) (cid:107) ∂ T h ≤ (( c − c j )( q n − jh − q njh ) , e q n jh ) T h + ( ∂ + t u njh − ∂ t Π jW u nj , e u n jh ) T h + (( β − β j ) · ∇ ( u n − jh − u njh ) , e u n jh ) T h − (cid:104) ( β − β j ) · n , ( u n − jh − u njh ) e (cid:98) u n jh (cid:105) ∂ T h = T + T + T . Next, we estimate { T i } i =1 . By the condition (2.1), there exist 0 < α < T = (( c − c j )( e q n − jh − ∆ t∂ + t q njh ) , e q n jh ) T h ≤ α (cid:16) (cid:107)√ ¯ ce q n jh (cid:107) T h + (cid:107)√ ¯ ce q n − jh (cid:107) T h (cid:17) + C ∆ t (cid:107) ∂ + t q njh (cid:107) T h ,T = ( ∂ + t ( u njh − Π jW u nj ) + ∂ + t Π jW u nj − ∂ t Π jW u nj , e u n jh ) T h ≤ C (cid:16) (cid:107) ∂ + t ( u njh − Π jW u nj ) (cid:107) T h + (cid:107) ∂ + t Π jW u nj − ∂ t Π jW u nj (cid:107) T h + (cid:107) e u n jh (cid:107) T h (cid:17) . To treat the term T , we use the technique in the proof of Lemma 6, where we treat the term R .For γ = β − β j , we have T ≤ (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,K (cid:107)∇ ( u n − jh − u njh ) (cid:107) K (cid:107) e u n jh (cid:107) K + (cid:88) K ∈T h (cid:107) γ − Π γ (cid:107) ∞ ,∂K (cid:107) u n − jh − u njh (cid:107) ∂K ( (cid:107) e (cid:98) u n jh − e u n jh (cid:107) ∂K + (cid:107) e u n jh (cid:107) ∂K )+ (cid:107) Π γ (cid:107) ∞ , T h (cid:107) ¯ ce q n jh (cid:107) T h (cid:107) u n − jh − u njh (cid:107) T h + (cid:107) (¯ c − c j ) Π γ (cid:107) ∞ , T h (cid:107) q n − jh − q njh (cid:107) T h (cid:107) u n − jh − u njh (cid:107) T h = T + T + T + T . For T , use the local inverse inequality: T ≤ C ( (cid:107) e u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t u njh (cid:107) T h + (cid:107) e u n jh (cid:107) T h ) .
14 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion EquationsApply the trace inequality and inverse inequality for the term T to give T ≤ C (cid:88) K ∈T h h K (cid:107) γ (cid:107) , ∞ ,K h − / K (cid:107) u n − jh − u njh (cid:107) K ( (cid:107) e (cid:98) u n jh − e u n jh (cid:107) ∂K + h − / K (cid:107) e u n jh (cid:107) K ) ≤ C ( (cid:107) e u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t u njh (cid:107) T h + (cid:107) e u n jh (cid:107) T h ) + 14 (cid:107)√ τ ( e (cid:98) u n jh − e u n jh ) (cid:107) ∂ T h . For the terms T and T , use Young’s inequality to obtain T ≤ − α (cid:107)√ ¯ ce q n jh (cid:107) T h + C ( (cid:107) e u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t u njh (cid:107) T h ) ,T ≤ − α (cid:107)√ ¯ ce q n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t q njh (cid:107) T h + C ( (cid:107) e u n − jh (cid:107) T h + ∆ t (cid:107) ∂ + t u njh (cid:107) T h ) . We add (3.15) from n = 1 to n = N , and use the above inequalities to getmax ≤ n ≤ N (cid:107) e u n jh (cid:107) T h + N (cid:88) n =1 (cid:107) e u n jh − e u n − jh (cid:107) T h + ∆ t N (cid:88) n =1 (cid:107)√ ¯ ce q n jh (cid:107) T h + (cid:107)√ τ ( e u n jh − e (cid:98) u n jh ) (cid:107) T h ≤ C ∆ t N (cid:88) n =1 (cid:107) e u n jh (cid:107) T h + C N (cid:88) n =1 (∆ t (cid:107) ∂ + t u njh (cid:107) T h + ∆ t (cid:107) ∂ + t q njh (cid:107) T h )+ C N (cid:88) n =1 (∆ t (cid:107) ∂ + t ( u njh − Π jW u nj ) (cid:107) T h + ∆ t (cid:107) ∂ + t Π jW u nj − ∂ t Π jW u nj (cid:107) T h )+ C (cid:107) e q jh (cid:107) T h + C (cid:107) e u jh (cid:107) T h . (3.16)Now we move to bound the terms on the right side of the above inequality as follows,∆ t N (cid:88) n =1 (cid:107) ∂ + t u njh (cid:107) T h = ∆ t N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ∂ t u njh dt (cid:21) ≤ C ∆ t (cid:107) ∂ t u njh (cid:107) L (0 ,T ; L (Ω)) , ∆ t N (cid:88) n =1 (cid:107) ∂ + t q njh (cid:107) T h = ∆ t N (cid:88) n =1 (cid:90) Ω (cid:20)(cid:90) t n t n − ∂ t q njh dt (cid:21) ≤ C ∆ t (cid:107) ∂ t q njh (cid:107) L (0 ,T ; L (Ω)) , ∆ t N (cid:88) n =1 (cid:107) ∂ + t ( u njh − Π jW u nj ) (cid:107) T h ≤ C (cid:107) ∂ t ( u jh − Π jW u j ) (cid:107) L (0 ,T ; L (Ω)) , ∆ t N (cid:88) n =1 (cid:107) ∂ + t Π jW u nj − ∂ t Π jW u nj (cid:107) T h ≤ C ∆ t (cid:107) ∂ tt Π jW u j (cid:107) L (0 ,T ; L (Ω)) . Gronwall’s inequality and the estimates above applied to (3.16) give the result.
The following element-by-element postprocessing is defined in [11]: Find u n(cid:63)jh ∈ P k +1 ( K ) such thatfor all ( z h , w h ) ∈ [ P k +1 ( K )] ⊥ × P ( K )( ∇ u n(cid:63)jh , ∇ z h ) K = − ( c j q njh , ∇ z h ) K , (3.17a)( u n(cid:63)jh , w h ) K = ( u h , w h ) K , (3.17b)where [ P k +1 ( K )] ⊥ = { z h ∈ P k +1 ( K ) | ( z h , K = 0 } .15. Chen, L. Pi, L. Xu Y. Zhang Lemma 9.
For any t ∈ [0 , T ] and k ≥
1, we have the following error estimate for the postprocessedsolution: (cid:107) Π k +1 u nj − u n(cid:63)jh (cid:107) T h ≤ C (cid:107) Π jW u nj − u njh (cid:107) T h + Ch (cid:107) c j ( q njh − q nj ) (cid:107) T h + Ch (cid:107)∇ ( u nj − Π k +1 u nj ) (cid:107) T h . Proof.
By the properties of Π W and Π k +1 , we obtain(Π jW u nj , w ) K = ( u nj , w ) K , for all w ∈ P ( K ) , (Π k +1 u nj , w ) K = ( u nj , w ) K , for all w ∈ P ( K ) . Hence, for all w ∈ P ( K ), we have(Π W u nj − Π k +1 u nj , w ) K = 0 . Let e njh = u n(cid:63)jh − u njh + Π jW u nj − Π k +1 u nj . Equation (3.17) and an inverse inequality give (cid:107)∇ e njh (cid:107) K = ( ∇ ( u n(cid:63)jh − u jh ) , ∇ e nj,h ) K + ( ∇ (Π jW u nj − Π k +1 u nj ) , ∇ e njh ) K = ( −∇ u njh − c j q njh , ∇ e njh ) K + ( ∇ (Π jW u nj − Π k +1 u nj ) , ∇ e njh ) K = ( ∇ (Π jW u nj − u njh ) − ( q njh − q nj ) + ∇ ( u nj − Π k +1 u nj ) , ∇ e njh ) K . This implies (cid:107)∇ e njh (cid:107) K ≤ C ( h − K (cid:107) Π jW u nj − u njh (cid:107) K + (cid:107) c j ( q njh − q nj ) (cid:107) K + (cid:107)∇ ( u nj − Π k +1 u nj ) (cid:107) K ) . (3.18)Since ( e h , K = 0, apply the Poincar´e inequality and the estimate (3.18) to give (cid:107) e njh (cid:107) K ≤ Ch K (cid:107)∇ e njh (cid:107) K ≤ C ( (cid:107) Π jW u nj − u njh (cid:107) K + h K (cid:107) c j ( q njh − q nj ) (cid:107) K + h K (cid:107)∇ ( u nj − Π k +1 u nj ) (cid:107) K ) . Hence, we have (cid:107) Π k +1 u nj − u n(cid:63)jh (cid:107) T h ≤ (cid:107) Π k +1 u nj − Π jW u nj − u n(cid:63)jh + u njh (cid:107) T h + (cid:107) Π jW u nj − u njh (cid:107) T h ≤ C (cid:107) Π jW u nj − u njh (cid:107) T h + Ch (cid:107) c j ( q njh − q nj ) (cid:107) T h + Ch (cid:107)∇ ( u nj − Π k +1 u nj ) (cid:107) T h . From Lemma 9 and the estimate in (2.4a) we complete the proof of (3.4) in Theorem 1.
In this section, we present some numerical tests of the Ensemble HDG method for parameterizedconvection diffusion PDEs. Although we derived the a priori error estimates for diffusion domi-nated problems, we also present numerical results for the convection dominated case to show theperformance of the Ensemble HDG method for the convection dominated diffusion problems. Forall examples, we take τ = 1 + max ≤ j ≤ J (cid:107) β j (cid:107) , ∞ so that (2.2) is satisfied, the coefficients c j satisfythe condition (2.1), and a group of simulations are considered containing J = 3 members. Let Eu j
16 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equationsbe the error bewteen the exact solution u j at the final time T = 1 and the Ensemble HDG solution u Njh , i.e., Eu j = (cid:107) u Nj − u Njh (cid:107) T h . Let E q j = (cid:118)(cid:117)(cid:117)(cid:116) ∆ t N (cid:88) n =1 (cid:107) q nj − q njh (cid:107) T h , and Eu (cid:63)j = (cid:118)(cid:117)(cid:117)(cid:116) ∆ t N (cid:88) n =1 (cid:107) u nj − u n(cid:63)jh (cid:107) T h . Example 1.
We first test the convergence rate of the Ensemble HDG method for diffusion domi-nated PDEs on a square domain Ω = [0 , × [0 , c = 0 . , c = 0 . , c = 0 . , β = 1 . y, x ] , β = 1 . y, x ] , β = 1 . y, x ] ,u j = sin( t ) sin( x ) sin( y ) /j, q j = − /c j ∇ u j , j = 1 , , , and the initial conditions, boundary conditions, and source terms are chosen to match the exactsolution of Equation (1.1).In order to confirm our theoretical results, we take ∆ t = h when k = 0 and ∆ t = h when k = 1.The approximation errors of the Ensemble HDG method are listed in Table 1 and the observedconvergence rates match our theory. 17. Chen, L. Pi, L. Xu Y. ZhangTable 1: History of convergence for Example 1.Degree h √ E q Eu Eu (cid:63) Error Rate Error Rate Error Rate k = 0 2 − − − − − k = 1 2 − − − − − h √ E q Eu Eu (cid:63) Error Rate Error Rate Error Rate k = 0 2 − − − − − k = 1 2 − − − − − h √ E q Eu Eu (cid:63) Error Rate Error Rate Error Rate k = 0 2 − − − − − k = 1 2 − − − − − Example 2.
Next, we perform Ensemble HDG computations for the convection dominated casewith exact solutions haveing interior layers. But we do not attempt to compute convergence rateshere; instead for illustration we plot all the ensemble members { u jh } j =1 at the final time T = 0 . , × [0 ,
1] and it is uniformly partition into 131072 triangles ( h = √ / t = 1 / u and right is u h computed by Ensemble HDG.Figure 2: Left is the exact solution u and right is u h computed by Ensemble HDG.to match Equation (1.1) for the data c = 10 , c = 2 × , c = 3 × , β = [2 , , β = [3 , , β = [4 , , and the exact solutions { u j } j =1 are chosen as u = sin( t ) x (1 − x ) y (1 − y )
12 + arctan 2 √ c (cid:16) − (cid:0) x − (cid:1) − (cid:0) y − (cid:1) (cid:17) π ,u = sin( t ) x (1 − x ) y (1 − y )
12 + arctan 2 √ c (cid:16) − (cid:0) x − (cid:1) − (cid:0) y − (cid:1) (cid:17) π ,u = sin( t ) x (1 − x ) y (1 − y )
12 + arctan 2 √ c (cid:16) − (cid:0) x − (cid:1) − (cid:0) y − (cid:1) (cid:17) π .
19. Chen, L. Pi, L. Xu Y. ZhangFigure 3: Left is the exact solution u and right is u h computed by Ensemble HDG.Figure 4: Left is solution u h and right is the postprocessed solution u (cid:63) h . Example 3.
Fianlly, we perform the Ensemble HDG method for a group of convection dominatedproblems without exact solutions. In this example, the problems exhibit not interior layers butboundary layers. It is well known that the boundary layers are more difficult than interior layersfor all numerical methods. Since in Example 2 the Ensemble HDG captured the interior layerswithout oscillations, we didn’t plot the postprocessed solutions there. However, our numerical testshows that the postprocessed solutions u (cid:63)jh are better than u jh for solutions with boundary layers;see e.g. Figures 4 to 6. We note there is no superconvergent rate even for a single convectiondominated diffusion problem PDE using HDG methods, see, e.g. [18].We plot all the ensemble members u jh and u (cid:63)jh at the final time T = 0 . c = 60 , c = 120 , c = 180 , β = [2 , , β = [3 , , β = [4 , ,f = 2 , f = 5 , f = 8 .
20 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion EquationsFigure 5: Left is solution u h and right is the postprocessed solution u (cid:63) h .Figure 6: Left is solution u h and right is the postprocessed solution u (cid:63) h .21. Chen, L. Pi, L. Xu Y. Zhang In this work, we first devised a superconvergent Ensemble HDG method for parameterized con-vection diffusion PDEs. This Ensemble HDG method shares one common coefficient matrix andmultiple RHS vectors, which is more efficient than performing separate simulations. We provedoptimal error estimates for the flux q j and the scalar variable u j ; moreover, we obtained the super-convergent rate for u j . As far as we are aware, this is the first time in the literature.There are a number of topics that can be explored in the future, including devising high ordertime stepping methods, a group of convection dominated diffusion PDEs, and stochastic PDEs. Acknowledgements
G. Chen is supported by China Postdoctoral Science Fou-ndation grant 2018M633339. G. Chen thanks Missouri University of Science and Technology forhosting him as a visiting scholar; some of this work was completed during his research visit. L. Xuis supported in part by a Key Project of the Major Research Plan of NSFC grant no. 91630205and the NSFC grant no. 11771068. The authors thank Dr. John Singler for his comments andedits, which significantly improved this paper.
In this section we only give a proof for (3.12a) and (3.12b), since the rest are similar. To prove(3.12c)-(3.12e), we differentiate the error equations in Lemma 10 with respect to time t . It is easyto check that the operators Π jW commute with the time derivative, i.e., ∂ t Π jW u j = Π jW ∂ t u j , sincethe velocity vector fields β j are independent of time t . Lemma 10.
We have the following equalities( c j Π jV q j , r h ) T h − (Π jW u j , ∇ · r h ) T h + (cid:104) P M u j , r h · n (cid:105) ∂ T h = ( c j ( Π jV q j − q j ) , r h ) T h , ( ∇ · Π jV q j , v h ) T h − (cid:104) Π jV q j · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW u j − P M u j ) , v h − (cid:98) v h (cid:105) ∂ T h +( β j · ∇ Π jW u j , v h ) T h − (cid:104) β j · n , Π jW u j (cid:98) v h (cid:105) ∂ T h = ( f j − ∂ t u j , v h ) T h , for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h .The proof is similar to the proof of Lemma 4, hence we omit it here.To simplify the notation, we set ε u j h = u jh − Π jW u j , ε q j h = q jh − Π jV q j , ε (cid:98) u j h = (cid:98) u jh − P M u j . Subtract (3.11) from (1.7) to get the following
Lemma 11.
We have the error equations( c j ε q jh , r h ) T h − ( ε ujh , ∇ · r h ) T h + (cid:104) ε (cid:98) ujh , r h · n (cid:105) ∂ T h = ( c j ( Π jV q j − q j ) , r h ) T h , (6.2a)( ∇ · ε q jh , v h ) T h − (cid:104) ε q jh · n , (cid:98) v h (cid:105) ∂ T h + ( β j · ∇ ε ujh , v h ) T h + (cid:104) τ ( ε ujh − ε (cid:98) ujh ) , v h − (cid:98) v h (cid:105) ∂ T h −(cid:104) β j · n , ε ujh (cid:98) v h (cid:105) ∂ T h = ( ∂ t u j − ∂ t Π jW u j , v h ) T h . (6.2b)for all ( r h , v h , (cid:98) v h ) ∈ V h × W h × M h . 22 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations q j Lemma 12.
We have (cid:107)√ c j ε q jh (cid:107) T h + (cid:107)√ τ ( ε ujh − ε (cid:98) ujh ) (cid:107) ∂ T h ≤ C (cid:107) ∂ t u j − ∂ t Π jW u j (cid:107) T h + C (cid:107) q j − Π jV q j (cid:107) T h + C (cid:107) ε ujh (cid:107) T h . Proof.
We take ( r h , v h , (cid:98) v h ) = ( ε q jh , ε ujh , ε (cid:98) ujh ) in (6.2), and add them together to get (cid:107)√ c j ε q jh (cid:107) T h + (cid:107)√ τ ( ε ujh − ε (cid:98) ujh ) (cid:107) T h + ( β j · ∇ ε ujh , ε ujh ) T h − (cid:104) β j · n , ε ujh ε (cid:98) ujh (cid:105) ∂ T h = ( c j ( Π jV q j − q j ) , ε q jh ) T h + ( ∂ t u j − ∂ t Π jW u j , ε ujh ) T h . By Green’s formula and the fact (cid:104) ( β j · n ) ε (cid:98) ujh , ε (cid:98) ujh (cid:105) ∂ T h = 0 we have( β j · ∇ ε ujh , ε ujh ) T h − (cid:104) β j · n , ε ujh ε (cid:98) ujh (cid:105) ∂ T h ≤ (cid:107) (cid:113) | β j · n | ( ε ujh − ε (cid:98) ujh ) (cid:107) ∂ T h . (6.3)Then by condition (2.2), we get the desired result. The next step is the consideration of the dual problems: c j Φ j + ∇ Ψ j = 0 in Ω , ∇ · Φ j − β j · ∇ Ψ j = Θ j in Ω , Ψ j = 0 on ∂ Ω . (6.4) Elliptic regularity.
To obatin the superconvergent rate, we are going to assume that the domainΩ is such that for any Θ j ∈ L (Ω), we have the regularity estimates for these boundary valueproblems (6.4): (cid:107) Φ j (cid:107) H (Ω) + (cid:107) Ψ j (cid:107) H (Ω) ≤ C (cid:107) Θ j (cid:107) L (Ω) . (6.5)It is well known that this holds whenever Ω is a convex polyhedral domain. Lemma 13.
If the elliptic regularity inequality (6.4) holds, then we have the error estimates (cid:107)√ c j ε q jh (cid:107) T h + (cid:107)√ τ ( ε ujh − ε (cid:98) ujh ) (cid:107) ∂ T h ≤ C A j , (cid:107) ε ujh (cid:107) T h ≤ Ch k +min { k, } A j , where A j = (cid:107) u j − Π jW u j (cid:107) T h + (cid:107) q j − Π jV q j (cid:107) T h + (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h . Proof.
Similar to Lemma 10, we have the following equations:( c j Π jV Φ j , r h ) T h − (Π jW Ψ , ∇ · r h ) T h + (cid:104) P M Ψ j , r h · n (cid:105) ∂ T h = ( c j ( Π jV Φ j − Φ j ) , r h ) T h , ( ∇ · Π jV Φ j , v h ) T h − (cid:104) Π jV Φ j · n , (cid:98) v h (cid:105) ∂ T h + (cid:104) τ (Π jW Ψ − P M Ψ j ) , v h − (cid:98) v h (cid:105) ∂ T h − ( β j · ∇ Π jW Ψ j , v h ) T h + (cid:104) β j · n , Π jW u j (cid:98) v h (cid:105) ∂ T h = (Θ j , v h ) T h .
23. Chen, L. Pi, L. Xu Y. ZhangTake ( r h , v h , (cid:98) v h ) = ( ε q jh , ε ujh , ε (cid:98) ujh ) and Θ j = ε ujh above to get, (cid:107) ε ujh (cid:107) T h = ( ∇ · Π jV Φ j , ε ujh ) T h − (cid:104) Π jV Φ j · n , ε (cid:98) ujh (cid:105) ∂ T h + (cid:104) τ (Π jW Ψ j − P M Ψ j ) , ε ujh − ε (cid:98) ujh (cid:105) ∂ T h − ( β j · ∇ Π jW Ψ j , ε ujh ) T h + (cid:104) β j · n , Π jW Ψ j ε (cid:98) ujh (cid:105) ∂ T h . By (6.2a) one gets (cid:107) ε ujh (cid:107) T h = ( c j ε q jh , Π jV Φ j ) T h − ( c j ( Π jV q j − q j ) , Π jV Φ j ) T h + (cid:104) β j · n , Π jW Ψ j ε (cid:98) ujh (cid:105) ∂ T h + (cid:104) τ (Π jW Ψ j − P M Ψ j ) , ε ujh − ε (cid:98) ujh (cid:105) ∂ T h − ( β j · ∇ Π jW Ψ j , ε ujh ) T h . Hence, (cid:107) ε ujh (cid:107) T h = (Π jW Ψ j , ∇ · ε q jh ) T h − (cid:104) P M Ψ j , ε q jh · n (cid:105) ∂ T h − ( c j ( Π jV Φ j − Φ j ) , ε q jh ) T h − ( c j ( Π jV q j − q j ) , Π jV Φ j ) T h + (cid:104) τ (Π jW Ψ j − P M Ψ j ) , ε ujh − ε (cid:98) ujh (cid:105) ∂ T h − ( β j · ∇ Π jW Ψ j , ε ujh ) T h + (cid:104) β j · n , Π jW Ψ j ε (cid:98) ujh (cid:105) ∂ T h . By Green’s formula one gets (cid:107) ε ujh (cid:107) T h = (Π jW Ψ , ∇ · ε q jh ) T h − (cid:104) P M Ψ j , ε q jh · n (cid:105) ∂ T h − ( c j ( Π jV Φ j − Φ j ) , ε q jh ) T h − ( c j ( Π jV q j − q j ) , Π jV Φ j ) T h + (cid:104) τ (Π jW Ψ − P M Ψ) , ε ujh − ε (cid:98) ujh (cid:105) ∂ T h + ( β j · ∇ ε ujh , Π jW Ψ j ) T h + (cid:104) β j · n , Π jW Ψ j ( ε (cid:98) ujh − ε ujh ) (cid:105) ∂ T h = (Π jW Ψ j , ∇ · ε q jh ) T h − (cid:104) P M Ψ j , ε q jh · n (cid:105) ∂ T h − ( c j ( Π jV Φ j − Φ j ) , ε q jh ) T h − ( c j ( Π jV q j − q j ) , Π jV Φ j ) T h + (cid:104) τ (Π jW Ψ j − P M Ψ j ) , ε ujh − ε (cid:98) ujh (cid:105) ∂ T h + ( β j · ∇ ε ujh , Π jW Ψ j ) T h − (cid:104) β j · n , ε ujh P M Ψ j (cid:105) ∂ T h + (cid:104) β j · n , ε ujh P M Ψ j (cid:105) ∂ T h + (cid:104) β j · n , Π jW Ψ j ( ε (cid:98) ujh − ε ujh ) (cid:105) ∂ T h . By (6.2b) one gets (cid:107) ε ujh (cid:107) T h = − ( c j ( Π jV Φ j − Φ j ) , ε q jh ) T h − ( c j ( Π jV q j − q j ) , Π jV Φ j ) T h + (cid:104) β j · n , ( ε ujh − ε (cid:98) ujh )( P M Ψ j − Π jW Ψ j ) (cid:105) ∂ T h + ( β j · ∇ Π jW Ψ j , Π jW u j − u j ) T h + ( ∂ t u j − ∂ t Π jW u j , Π jW Ψ j ) T h = (cid:88) i =1 R i .
24 Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion EquationsWe estimate { R i } i =1 term by term: R ≤ Ch (cid:107) Φ j (cid:107) (cid:107) ε q jh (cid:107) T h ≤ Ch (cid:107) ε ujh (cid:107) T h + Ch ( (cid:107) u j − Π jW u j (cid:107) T h + (cid:107) q j − Π jV q j (cid:107) T h ) (cid:107) ε ujh (cid:107) T h ,R ≤ Ch min { k, } (cid:107) Φ j (cid:107) (cid:107) q j − Π jV q j (cid:107) T h ≤ Ch min { k, } (cid:107) ε ujh (cid:107) T h (cid:107) q j − Π jV q j (cid:107) T h ,R ≤ Ch +min { k, } (cid:107) Ψ j (cid:107) (cid:107)√ τ ( ε ujh − ε (cid:98) ujh ) (cid:107) ∂ T h ≤ Ch +min { k, } ( (cid:107) u j − Π jW u j (cid:107) T h + (cid:107) q j − Π jV q j (cid:107) T h ) (cid:107) ε ujh (cid:107) T h + Ch +min { k, } (cid:107) ε ujh (cid:107) T h ,R = (( β j − Π β j ) · ∇ Π jW Ψ j , Π jW u j − u j ) T h ≤ Ch (cid:107) Ψ j (cid:107) (cid:107) u j − Π jW u j (cid:107) T h ≤ Ch (cid:107) ε ujh (cid:107) T h (cid:107) u j − Π jW u j (cid:107) T h ,R ≤ Ch min { k, } (cid:107) Ψ j (cid:107) (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h ≤ Ch min { k, } (cid:107) ε ujh (cid:107) T h (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h . Hence, we have (cid:107) ε ujh (cid:107) T h ≤ Ch min { k, } (cid:16) (cid:107) u j − Π jW u j (cid:107) T h + (cid:107) q j − Π jV q j (cid:107) T h + (cid:107) ∂ t u j − Π jW ∂ t u j (cid:107) T h (cid:17) . References [1] A. Buffa, T. J. R. Hughes, and G. Sangalli. Analysis of a multiscale discontinuous Galerkinmethod for convection-diffusion problems.
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