Generalized DG-Methods for Highly Indefinite Helmholtz Problems based on the Ultra-Weak Variational Formulation

نویسندگان

  • J. M. Melenk
  • A. Parsania
  • S. Sauter
چکیده

We develop a stability and convergence theory for the Ultra Weak Variational Formulation (UWVF) of a highly indefinite Helmholtz problem in Rd, d ∈ {1, 2, 3}. The theory covers conforming as well as nonconforming 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 UWVF admits a unique solution under much weaker conditions. As an application we present the error analysis for the hp-version of the finite element method explicitly in terms of the mesh width h, polynomial degree p and wave number k. It is shown that the optimal convergence order estimate is obtained under the conditions that kh/ √ p is sufficiently small and the polynomial degree p is at least O(log k). AMS Subject Classifications: 35J05, 65N12, 65N30

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تاریخ انتشار 2012