A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

Abstract

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H1 and L2 norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.

Weak Galerkin finite element method for linear poroelasticity problems

“…In analogy with discrete weak gradient, discrete weak divergence is introduced in [31] where the proposed WG mixed finite element method is applicable for general finite element partitions consisting of shape regular polygons in 2D or polyhedra in 3D. At present, WG method has been widely used to solve various partial differential equations, such as Helmholtz equations [19,9,33], linear parabolic equations [17,37,36], Biharmonic problems [18,7], Stokes problems [22], stochastic partial differential equations [40,41], and heat equations [39,38] etc.…”

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

“…Similarly, the WGMFEM with a stabilization term is introduced in [31] based on the definition of discrete weak divergence operator and applicable for general finite element partitions consisting of regular polygon shapes in 2D or polyhedra in 3D. Moreover, WGMFEM is developed in second-order elliptic equations with Robin boundary conditions [32], parabolic differential equations with memory [33], Helmholtz equations with large wave numbers [34], and biharmonic equations [35]. In addition to solving deterministic problems, there has also been some progress in the WG method for stochastic problems, such as stochastic Brinkman problems [36], elliptic problems with stochastic jump coefficients [37], and random interface grating problems [38].…”

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

“…In recent years, many numerical methods have been developed to solve and analyze the Helmholtz equation, for instance, finite difference method [2][3][4][5], conforming finite element method [6,7], boundary element method [8], weak Galerkin finite element method [9][10][11][12], spectral method [13], and adaptive finite element method [14] . It is also well known that the discontinuous Galerkin methods are flexible and highly parallelizable, and hence discontinuous Galerkin methods are widely used to solve the Helmholtz equation numerically, such as interior penalty discontinuous Galerkin method [15], hybridizable discontinuous Galerkin method [16,17], local discontinuous Galerkin method [18], and the references therein.…”