Weak Galerkin mixed finite element methods for parabolic equations with memory

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both \begin{document}$ |\|·|\| $\end{document} and \begin{document}$ L^2 $\end{document} norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces \begin{document}$ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $\end{document} for \begin{document}$ k = 0 $\end{document} and the numerical convergence rates confirm the theoretical results.


1.
Introduction. Weak Galerkin mixed finite element method (WG-MFEM), which can be viewed as an extension of weak Galerkin finite element method [7], was first proposed by Wang and Ye for the second-order elliptic equations [8]. WG-MFEM can provide accurate numerical approximations for both the primary and flux variables just as the existing mixed finite element methods. In particular, based on a use of weak divergence, WG-MFEM is designed to be more flexible in using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. Later on, WG-MFEM was successfully applied to the biharmonic equations [3]. In [4], a hybridized formulation for the WG-MFEM was proposed which can reduce significantly the discrete linear system. For secondorder elliptic equations with Robin boundary condition, Zhang et al. studied the WG-MFEM by proving inf-sup condition with a suitable norm and testing a series of numerical examples [11]. Furthermore, WG-MFEMs for heat equations are developed in [12] by Zhou et al. However, study on WG-FEMs for the parabolic equations with memory is still limited.
Compared to the traditional heat equation, the parabolic differential equation with integral term is more suitable in modeling the heat conduction in materials with memory [10], which is also known as parabolic integro-differential equation. Due to the integral term, the analysis and implementation of numerical schemes for parabolic integro-differential equation are more complex accordingly. In order to obtain both the primary and flux unknowns, several mixed finite element methods have been developed for the generalized equation in recent years [1,2,5,6,13,14,15]. In this paper, we consider the initial-boundary value problem for linear parabolic equation with memory term where Ω ⊂ R 2 is a bounded convex polygonal domain with boundary ∂Ω, the time interval J = (0, T ] with T > 0, A = A(x) and B = B(x) are sufficiently smooth functions. In view of the accuracy and the flexibility of the finite element partition of WG-MFEM, we present WG-MFEM approaches for the linear parabolic integrodifferential problem (1).
The main goal of this paper is to extend the WG-MFEM to the parabolic integrodifferential equation with the aid of discrete weak divergence operator. We will prove the existence and uniqueness of the WG-MFEM solutions. In the error analysis, to obtain the optimal convergence order of the WG-MFEMs, we will propose a generalized weak Galerkin mixed elliptic projection corresponding the mixed variational form of the integro-differential equation.
The rest of paper is organized as follows. In Section 2, we present a semidiscrete and a backward Euler fully discrete WG-MFEM for the parabolic integro-differential problem (1). In Section 3, some useful lemmas are listed. We prove the existence and uniqueness of the WG-MFEM approximations in Section 4. An appropriate generalized weak Galerkin mixed elliptic projection is introduced in Section 5. Optimal error estimates in both | · | and L 2 norms of the semidiscrete scheme and the fully discrete scheme are established in Section 6. In Section 7, numerical experiments are carried out for two examples to verify the theoretical results. Finally, we draw our conclusions in Section 8.

2.
Weak Galerkin mixed finite element schemes. In the parabolic integrodifferential problem (1), we define a flux variable q = −A∇u − t 0 B∇udτ and let α = A −1 , β = αB, γ = −∇β. Then problem (1) can be rewritten as the following system of first order partial differential equations in with boundary and initial conditions The variational weak formulation of this system is to seek {q, u} : J → H(div; Ω) × L 2 (Ω) such that for any t ∈ J,

WEAK GALERKIN MIXED FEMS FOR PARABOLIC EQUATIONS WITH MEMORY 515
where (·, ·) is the standard L 2 -inner product. We recall some notations and definitions for the WG-MFEMs used in [8] and [11]. Let T h = {T } be a quasi-uniform triangulation of Ω with h = max T ∈T h diamT . Denote by E h the collection of all edges in T h and E 0 h = E h \∂Ω. For any element T ∈ T h , we denote the space of weak vector-valued function on T by where n is the unit outward normal direction of T on ∂T . Let M = T ∈T h M (T ) be the weak vector-valued function space on T h , and V be a subspace of M as For a non-negative integer k, we denote P k (T ) the set of polynomials on T with degree no more than k. By using a combination of sets {[P k (T )] 2 , P k (e), P k+1 (T )}, we define a finite element space W h as and a finite element space V h ⊂ V as where the unit vector n e is normal to e for e ∈ E 0 h , and outward normal to e for e ∈ E h \E 0 h . Then we introduce the discrete weak divergence operator ∇ w,k · as an approximation to ∇· in finite element spaces [11]. For each v ∈ V , the discrete weak divergence (∇ w,k · v)| T ∈ P k (T ) on element T is determined by Thus we have the discrete weak divergence operator ∇ w,k+1 · : V h → W h . Considering the variational form (5), we define bilinear operators as To ensure the weak Galerkin mixed finite element scheme has a unique solution, a stabilized form a s (·, ·) is defined by where Now we get a semidiscrete weak Galerkin mixed finite element method (SWG-MFEM) for problem (2)(3)(4) with appropriately chosen initial values R h q(·, 0) and E h u(·, 0) to be defined in (51).

XIAOMENG LI, QIANG XU AND AILING ZHU
Let ∆t = T /N with positive integer N and t n = n∆t for n = 0, 1, · · · , N . We also obtain a backward Euler fully discrete weak Galerkin mixed finite element method 3. L 2 projections and some lemmas.
where Q 0 is the L 2 projection from [L 2 (T )] 2 to [P k (T )] 2 and Q b is the L 2 projection from L 2 (e) onto P k (e). Let Q h be the L 2 projection from L 2 (Ω) onto W h . In addition, we introduce a new norm | · | in finite element space V h and a norm · 1,h in the space W h + H 1 (Ω), which are defined by [11] | v| 2 = where [w] means the jump of w on edge e. It is easy to prove that a s (·, ·) is bounded and coercive, i.e., there exist positive constants C 1 , C 2 such that for any We recall some useful lemmas in [8] and [9].
where C 3 , C 4 are positive constants independent of h.
where ∇ h w is the gradient of w taken element-by-element.

WEAK GALERKIN MIXED FEMS FOR PARABOLIC EQUATIONS WITH MEMORY 517
Lemma 3.5. [8] Let u ∈ H k+2 (Ω) and q ∈ [H k+1 (Ω)] 2 be two smooth functions on Ω. Then we have 4. Existence and uniqueness of numerical schemes. In this section, we prove in Theorem 4.1 and 4.2 that the two numerical schemes defined in (15) and (16) are uniquely solvable.
Proof. We only need to prove that the following system of the corresponding homogenous equations of (15) has only trivial solution: for t ∈ J. First, we write (34) as It follows from the boundedness of a s (·, ·) and Lemma 3.1 that Due to Gronwall inequality, we have Let w = u h in (33) and v = q h in (34), then

XIAOMENG LI, QIANG XU AND AILING ZHU
By using (21) (38) and Lemma 3.1, we arrive at Integrating (40) from 0 to t with u h (·, 0) = 0, then we get By using Gronwall inequality again, we have which implies that q h = 0 and u h = 0 for any t ∈ J. Therefore, The SWG-MFEM (15) has a unique solution.
Proof. Similarly, we consider the following homogenous linear system (43) and v = q n h in (44). Then we have Notice that From (21) and (46), we estimate that Adding the equation above with n from 1 to m (1 < m ≤ N ) with u 0 h = 0, we arrive at Thus we get with the discrete Gronwall inequality. This implies that q n h = 0 and u n h = 0 for 1 ≤ n ≤ N . Therefore, The FWG-MFEM (16) has a unique solution.

5.
A generalized weak Galerkin mixed elliptic projection. In the study of finite element methods of differential equations of evolution, an elliptic projection associated the problems is usually introduced. The elliptic projection usually plays a key role in obtaining the optimal convergence order of finite element approximations. Since the integro-differential problem has a integral term, the projections for finite element methods solving differential equations (e.g. heat equation) will not work. Assume that q and u are the solutions of variational form (5). Here we introduce a generalized weak Galerkin mixed elliptic projection for the WG-MFEMs applicable to the mixed system (5) by defining a map First, we prove the existence and uniqueness of the solution (51). It suffices to show that the associated homogeneous system has only trivial solution, which is for any w ∈ W h , v ∈ V h We choose w = ∇ w · R h q and v = R h q in (52), then get ∇ w · R h q = 0 and which leads to From (20) and Lemma 3.1, we have Then we arrive at

XIAOMENG LI, QIANG XU AND AILING ZHU
A combination of (54) and (56) gives According to Gronwall inequality and inequality (54), we have E h u 1,h = 0, |||R h q||| = 0, respectively. Therefore the system (51) is uniquely solvable. This shows that the generalized weak Galerkin mixed elliptic projection is well defined. Next, we derive the error equations of R h q and E h u. Set On one hand, applying Lemma 3.2 to the first equation of (51), we have On the other hand, we have (αq, v 0 ) = a s (q, v) from the definition of s(·, ·). Notice that Q h (βu) = βQ h u. According to Lemma 3.3, we obtain Combining (59) with the second equation of (51), we have According to (58) and (60), we rewrite (51) as

WEAK GALERKIN MIXED FEMS FOR PARABOLIC EQUATIONS WITH MEMORY 521
are the solutions of (5) and {R h q, E h u} satisfies system (51), then we have Proof. Similar to the derivation of (37), it follows from the second equation of (61) that From Lemma 3.4 and 3.5, we get Notice that θ = η + Q h u − u. Combining (63) and (64), we have Using Gronwall inequality, we have Substitute w = η and v = ξ into (61), then Using equations (21) (64), Lemma 3.1 and Lemma 3.5, we obtain From ε-inequality and (66), we have Thus, combining (66) and the inequality above, we obtain

WEAK GALERKIN MIXED FEMS FOR PARABOLIC EQUATIONS WITH MEMORY 523
Next, from Lemma 3.7, we have and By Lemma 3.5 and 3.6, we get In summary, we have From (73), we rewrite (85) as It follows from Gronwall inequality and Theorem 5.1 that This completes the proof.
We take the derivatives with respect to t of the results in Theorem 5.1 and 5.2 respectively. Then the estimates of | ξ t | and η t are obtained in the following theorem.
Proof. Choose w ∈ W h in the first equation of (5) and notice that (Q h u t , w) = (u t , w), we obtain Subtracting (92) from the first equation of semidiscrete scheme (15) and considering (51), we get i.e., Choose v = {v 0 , v b } ∈ V h in the second equation of the (5), then subtract it from the second equation of (15), we get By Adding (94) and (95) together with w = η, v = ξ, we arrive at Notice the (21) and Lemma 3.1, we have By (38), We integrate (97) from 0 to t to arrive at where η(·, 0) = u h (·, 0) − E h u(·, 0) = 0. It follows from Gronwall inequality and Theorem 5.3 that By taking a derivative of (95) with respect to t, we get Choose w = η t in (94) and v = ξ in (100) respectively. By adding them together, we have Integrating (102) from 0 to t and using (38), we get where ξ(·, 0) = q h (·, 0) − R h q(·, 0) = 0. Finally, using Gronwall inequality and Theorem 5.3, we have for t ∈ J. This completes the proof.
, then there exists a positive constant C independent of h and ∆t such that Proof. We choose w ∈ W h in the first equation of (5) at t = t n to get (u t (·, t n ), w) + (∇ · q(·, t n ), w) = (f (·, t n ), w), 1 ≤ n ≤ N.
Subtracting (107) from the first equation of the problem (16), we have from (51) that Notice that (u t (·, t n ), w) = (Q h u t (·, t n ), w). By adding and subtracting δ t Q h u t (t n ) in the equation (108), we write (108) as . Similar to the deduction of (95), we get for 1 ≤ n ≤ N . We add (109) and (110) together with w = η n , v = ξ n to arrive at (δ t η n , η n ) + a s (ξ n , ξ n ) =(z n , η n ) − (δ t η n , η n ) + ∆t for 1 ≤ n ≤ N . We estimate each term on both side of the equation above as follows Similarly, we also have By Lemma 3.1 and (38), we obtain We substitute the above inequality into (111) and multiply it by 2∆t, then sum over n to obtain where η 0 = 0. Using discrete Gronwall inequality and Theorem 5.3, we obtain Next, if the backward difference operator δ t is applied to (110), we can get for 1 ≤ n ≤ N . Choosing w = δ t η n in (109) and v = ξ n in (121), we arrive at (δ t η n , δ t η n ) + a s (δ t ξ n , ξ n ) =(z n , δ t η n ) − (δ t ρ n , δ t η n ) =δ t ∆t where actually a s (δ t ξ n , ξ n ) = 1 2 δ t a s (ξ n , ξ n ) + ∆t 2 a s (δ t ξ n , δ t ξ n ) (a s (ξ n , ξ n ) − a s (ξ n−1 , ξ n−1 )), Then we substitute them into (111), multiply it by 2∆t and sum over n to obtain Notice that ξ 0 = 0. By applying the conclusion of (115) and (116), we finally get According to Theorem 5.1 and triangle inequality, the proof is completed.
7. Numerical experiments. In this section, the convergence rate of FWG-MFEM (16) by solving the integro-differential problems with two examples is tested. We take k = 0 in the discretization of the weak Galerkin mixed finite element spaces.
In each example, we set Ω = (0, 1) × (0, 1), J = (0, 1] and use the uniform triangular mesh shown in Figure 1. Denote e h = q N h − Q h q(·, N ) = {e 0 , e b } and ε h = u N h − Q h u(·, N ), which are measured by the following norms: The source term f can be obtained by substituting the exact u into the integodifferential equation.
In the first example, we set the coefficient A = B = 1 and the exact solution of the integro-differential problem (1) u = t 2 xy(1 − x)(1 − y).  In the second example, we set variable coefficients A = 1+x 2 +2y 2 , B = 1+2x 2 +y 2 , and the exact solution u = e −t sin(πx) sin(πy). Table 1-2 report the errors and convergence rate of the FWG-MFEM for solving the parabolic integro-differential equation in both examples with k = 0 respectively. As we can see, the weak Galerkin mixed finite element method has good performance for solving the parabolic integro-differential equations. For k = 0, the | · | -norm for e h and · -norm for ε h are of order O(∆t + h) and O(∆t + h 2 ) respectively. This verifies our theoretical analysis in Section 6. 8. Conclusions. In consideration of the practical requirement of accurate approximations to both primary and flux variables in the parabolic differential equations with memory term, we developed the semidiscrete and the backward Euler fully discrete weak Galerkin mixed finite element method with the aid of discrete weak divergence operator. The existence and uniqueness of the two weak Galerkin schemes were proved. We proposed a generalized weak Galerkin mixed elliptic projection corresponding the mixed variational form of the integro-differential equation, based on which, the optimal convergence order of the semidiscrete and fully discrete weak Galerkin mixed finite element methods in both | ·| and L 2 norms were established. The numerical experiments verified the accuracy of the weak Galerkin schemes studied in this paper.