A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers. (English) Zbl 1407.65302
Summary: 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 \(H^1\) and \(L^2\) norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
discrete weak divergence; Helmholtz equation; mixed finite element methods; weak Galerkin finite element methodsReferences:
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