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An adaptive multiresolution interior penalty discontinuous Galerkin method for wave equations in second order form. (English) Zbl 1460.74080
Summary: In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) discontinuous Galerkin (DG) methods for simulating scalar wave equations in second order form in space. The two key ingredients of the schemes include an interior penalty DG formulation in the adaptive function space and two classes of multiwavelets for achieving multiresolution. In particular, the orthonormal Alpert’s multiwavelets are used to express the DG solution in terms of a hierarchical structure, and the interpolatory multiwavelets are further introduced to enhance computational efficiency in the presence of variable wave speed or nonlinear source. Some theoretical results on stability and accuracy of the proposed method are presented. Benchmark numerical tests in 2D and 3D are provided to validate the performance of the method.
74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
74J05 Linear waves in solid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Overture; p4est
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