A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

Abstract

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L∞(L2) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L∞(L2) norm.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

“…Numerical methods applied for hyperbolic interface problems based on finite element framework can be mainly grouped by conforming FEM, discontinuous Galerkin (DG) and immersed FEMs. Finite element approximations of wave equation with interfaces via interface fitted conforming finite element algorithms are carried out in [12,14,15]. To avoid interface fitted mesh, immersed FEMs have been proposed to allow the interface to cut through elements so that simple structured Cartesian meshes can be employed that are not necessarily body-fitted (cf.…”

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

“…The performance of such kind of interface‐fitted FEMs depends not only on the quality of underlying finite element partition but also on the variational formulation of the problem. While the flux discontinuity of the solution can be captured in a variational formulation, the discontinuity of the solutions neither fit in the variational formulation nor satisfied in classical FEM solution spaces (see , and references therein). Many efforts have been made to develop alternative FEMs based on unfitted meshes for solving interface problems.…”