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A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers
Author(s) -
Wang Ruishu,
Wang Xiaoshen,
Zhai Qilong,
Zhang Kai
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22242
Subject(s) - mathematics , discontinuous galerkin method , helmholtz equation , finite element method , galerkin method , polygon mesh , piecewise , mathematical analysis , mixed finite element method , stability (learning theory) , boundary value problem , geometry , physics , computer science , thermodynamics , machine learning
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.