SIAM Journal on Numerical Analysis
Computational aspects of the ultra-weak variational formulation
Journal of Computational Physics
Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number
SIAM Journal on Numerical Analysis
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
Journal of Computational Physics
Solving Maxwell's equations using the ultra weak variational formulation
Journal of Computational Physics
Discontinuous Galerkin methods with plane waves for time-harmonic problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation
SIAM Journal on Numerical Analysis
Error Estimates of the Finite Element Method for Interior Transmission Problems
Journal of Scientific Computing
General DG-Methods for Highly Indefinite Helmholtz Problems
Journal of Scientific Computing
FEM with Trefftz trial functions on polyhedral elements
Journal of Computational and Applied Mathematics
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Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255-299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121-136], and plane wave approximation theory.