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MATHEMATICS OF COMPUTATION Volume 83, Number 289, September 2014, Pages 2101–2126 S 0025-5718(2014)02852-4 Article electronically published on May 5, 2014

A WEAK GALERKIN MIXED FINITE ELEMENT METHOD

FOR SECOND ORDER ELLIPTIC PROBLEMS

JUNPING WANG AND XIU YE

Abstract. A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using dis- continuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables.

Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical compu- tation. Optimal order error estimates in both discrete H1 and L2 norms are established for the corresponding weak Galerkin mixed finite element solutions.

1. Introduction

Weak Galerkin (WG) refers to a finite element technique for partial differential equations in which differential operators are approximated by their weak forms as distributions. In [29,30], a weak Galerkin method was introduced and analyzed for second order elliptic equations and the Stokes equation based on weak gradients and weak divergence. In this paper, we shall develop a new weak Galerkin method for second order elliptic equations formulated as a system of two first order linear equations. Our model problem seeks a flux function q = q(x) and a scalar function u = u(x) defined in an open bounded polygonal or polyhedral domain Ω ⊂ Rd (d = 2, 3) satisfying

αq+∇u = 0, ∇ · q = f, in Ω(1.1)

and the following Dirichlet boundary condition

(1.2) u = −g on ∂Ω,

where α = (αij(x))d×d ∈ [L∞(Ω)]d 2

is a symmetric, uniformly positive definite matrix on the domain Ω. A weak formulation for (1.1)-(1.2) seeks q ∈ H(div,Ω)

Received by the editor March 20, 2012 and, in revised form, November 23, 2012 and December 11, 2012.

2010 Mathematics Subject Classification. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50.

Key words and phrases. Weak Galerkin, finite element methods, discrete weak divergence, second order elliptic problems, mixed finite element methods.

The research of the first author was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

This research was supported in part by National Science Foundation Grant DMS-1115097.

c©2014 American Mathematical Society

2101

Licensed to Univ of Pittsburgh. Prepared on Fri Mar 4 12:28:22 EST 2016 for download from IP 136.142.124.50.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

http://www.ams.org/mcom/ http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-2014-02852-4

2102 JUNPING WANG AND XIU YE

and u ∈ L2(Ω) such that

(αq,v)− (∇ · v, u) = 〈g,v · n〉∂Ω, ∀v ∈ H(div,Ω),(1.3) (∇ · q, w) = (f, w), ∀w ∈ L2(Ω).(1.4)

Here L2(Ω) is the standard Hilbert space of square integrable functions on Ω, ∇·v is the divergence of vector-valued functions v on Ω, H(div,Ω) is the Sobolev space consisting of vector-valued functions v such that v ∈ [L2(Ω)]d and ∇ · v ∈ L2(Ω), (·, ·) stands for the L2-inner product in L2(Ω), and 〈·, ·〉∂Ω is the inner product in L2(∂Ω).

Galerkin methods based on the weak formulation (1.3)-(1.4) and finite dimen- sional subspaces of H(div,Ω) × L2(Ω) with piecewise polynomials are known as mixed finite element methods (MFEM). MFEMs for (1.1)-(1.2) treat q and u as unknown functions and are capable of providing accurate approximations for both unknowns [1, 3, 7–11, 27, 28]. All the existing MFEMs in literature possess local mass conservation that makes MFEM a competitive numerical technique in many applications such as oil reservoir and groundwater flow simulation in porous me- dia. On the other hand, MFEMs are formulated in subspaces of H(div,Ω)×L2(Ω) which requires a certain continuity of the finite element functions for the flux vari- able. More precisely, the flux functions must be sufficiently continuous so that the usual divergence is well-defined in the classical sense in L2(Ω). This continuity as- sumption in turn imposes a strong restriction on the structure of the finite element partition and the piecewise polynomials defined on them [9–11,27, 30].

The weak Galerkin method introduced in [29] (see also [22, 23] for extensions) was based on the use of weak gradients in the following variational formulation: find u ∈ H1(Ω) such that u = −g on ∂Ω and

(1.5) (α−1∇u,∇φ) = (f, φ), ∀φ ∈ H10 (Ω),

whereH1(Ω) is the Sobolev space consisting of functions for which all partial deriva- tives up to order one are square integrable, H10 (Ω) is the subspace of H

1(Ω) con- sisting of functions with vanishing value on ∂Ω. Specifically, the weak Galerkin finite element formulation in [22, 23, 29, 30] can be obtained from (1.5) by simply replacing the gradient ∇ by a discrete gradient ∇w (it was denoted as ∇d in [29]) defined by a distributional formula. The discrete weak gradient operator ∇w is locally-defined on each element. It has been demonstrated [24–26, 29, 30] that the weak Galerkin method enjoys an easy-to-implement formulation that is parameter free and inherits the physical property of mass conservation locally on each element. Furthermore, the weak Galerkin method has the flexibility of using discontinuous finite element functions, as was commonly employed in discontinuous Galerkin and hybridized discontinuous Galerkin methods [2, 15].

The goal of this paper is to extend the weak Galerkin method of [22, 23, 29] to the variational formulation (1.3)-(1.4) by following the idea of weak gradients. It is clear that divergence is the principle differential operator in (1.3)-(1.4). Thus, an essential part of the extension is the development of a weakly-defined discrete divergence operator, denoted by (∇w·), for a class of vector-valued weak functions in a finite element setting. Assuming that there is such a discrete divergence operator (∇w·) defined on a finite element space Vh for the flux variable q, then formally one would have a WG method for (1.1)-(1.2) that seeks qh ∈ Vh and uh ∈ Wh

Licensed to Univ of Pittsburgh. Prepared on Fri Mar 4 12:28:22 EST 2016 for download from IP 136.142.124.50.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A WEAK GALERKIN MFEM FOR SECOND ORDER ELLIPTIC PROBLEMS 2103

satisfying

(αqh,v)− (∇w · v, uh) = 〈g,v · n〉∂Ω, ∀v ∈ Vh, (∇w · qh, w) = (f, w), ∀w ∈ Wh,

where Wh is a properly defined finite element space for the scalar variable. The rest of the paper will provide details for a rigorous interpretation and justification of the above formal WG method, which shall be called WG-MFEM methods.

The WG-MFEM have the following features. First of all, the finite element partition of the domain Ω is allowed to consist of arbitrary shape of polygons for d = 2 and polyhedra for d = 3. Second, the flux approximation space Vh consists of two components, where the first one is given by piecewise polynomials on each polygon/polyhedra and the second is given by piecewise polynomials on the edges/faces of the polygon/polyhedra. The second component shall be used to approximate the normal component of the flux variable q on each edge/face. Furthermore, the scalar approximation space Wh consists of piecewise polynomials on each polygon/polyhedra with one degree higher than that of the flux. For example, the lowest order of such elements would consist of piecewise constant for flux and its normal component on each edge/face plus piecewise linear function for the scalar variable on each polygon/polyhedra. There is no continuity required for any of the finite element functions in WG-MFEM.

One close relative of the WG-MFEM is the mimetic finite difference (MFD) method, see [4–6,12] and the references cited therein. Both WG-MFEM and MFD share the same flexibility of using polygonal/polyhedral elements of the domain. The lowest order MFD approximates the flux by using only piecewise constants on each edge/face, and the scalar variable by using another piecewise constant function on each polygonal/polyhedral element, while WG-MFEM provides a wide class of numerical schemes with arbitrary order of polynomials. The arbitrary-order mimetic scheme of [4] is based on the philosophy of approximating the forms in (1.5) using nodal-based polynomials, which has very minimal overlap with the WG- MFEM. It should be pointed out that there are other numerical methods designed on general polygonal meshes in existing literature [2, 15, 17–20]. In particular, a comparison with existing numerical schemes should be conducted on a certain set of benchmark problems as demonstrated in [20]. More benchmarks can be found in [19].

Allowing arbitrary shape for mesh elements provides a convenient flexibility in both numerical approximation and mesh generation, especially in regions where the domain geometry is complex. Such a flexibility is also very much appreciated in adaptive mesh refinement methods. This is a highly desirable feature in practice since a single type of mesh technology is too restrictive in reso