Name
Abstract | ||
|---|---|---|
We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $$\mathbb R ^{d}$$ R d , $$d\in \{1,2,3\}$$ d 驴 { 1 , 2 , 3 } . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $$hp$$ hp -version of the finite element method explicitly in terms of the mesh width $$h$$ h , polynomial degree $$p$$ p and wavenumber $$k$$ k . It is shown that the optimal convergence order estimate is obtained under the conditions that $$kh/\sqrt{p}$$ kh / p is sufficiently small and the polynomial degree $$p$$ p is at least $$O(\log k)$$ O ( log k ) . On regular meshes, the first condition is improved to the requirement that $$kh/p$$ kh / p be sufficiently small. |
Year | DOI | Venue |
|---|---|---|
2013 | 10.1007/s10915-013-9726-8 | J. Sci. Comput. |
Keywords | Field | DocType |
Helmholtz equation at high wavenumber,Stability,Convergence,Discontinuous Galerkin methods,Ultra-weak variational formulation,Polynomial hp-finite elements | Discontinuous Galerkin method,Convergence (routing),Mathematical optimization,Polygon mesh,Mathematical analysis,Wavenumber,Galerkin method,Helmholtz free energy,Degree of a polynomial,Finite element method,Mathematics | Journal |
Volume | Issue | ISSN |
57 | 3 | 0885-7474 |
Citations | PageRank | References |
12 | 0.76 | 11 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| J. M. Melenk | 1 | 74 | 6.52 |
| A. Parsania | 2 | 12 | 0.76 |
| Stefan Sauter | 3 | 87 | 6.98 |
| JM Melenk | 4 | 12 | 0.76 |