Degree and wavenumber [in]dependence of Schwarz preconditioner for the DPG method

This note describes an implementation of a discontinuous Petrov Galerkin (DPG) method for acoustic waves within the framework of high order finite elements provided by the software package NGSolve. A technique to impose the impedance boundary condition weakly is indicated. Numerical results from this implementation show that a multiplicative Schwarz algorithm, with no coarse solve, provides a p-preconditioner for solving the DPG system. The numerical observations suggest that the condition number of the preconditioned system is independent of the frequency k and the polynomial degree p. 1 A Petrov Galerkin formulation We consider the Helmholtz equation modeling time harmonic acoustic waves in a homogenous medium, −∆u− k2u = f on Ω (1a) u = 0 on ∂Ω , (1b) Here we have considered the simplest Dirichlet boundary condition (postponing the discussion of impedance boundary condition to later), and Ω is a polygonal (2D) or polyhedral (3D) domain, partitioned into a simplicial finite element mesh Ωh. When k2 is not an eigenvalue of −∆ , this problem has a unique solution. We want to study its approximation by the so-called primal discontinuous Petrov Galerkin Jay Gopalakrishnan Portland State University, PO Box 751, Portland OR 97207, USA, e-mail: [email protected] Joachim Scöberl Wiedner Hauptstraße 8-10, TU Wien, 1040 Wein, Austria, e-mail: joachim.schoeberl@ tuwien.ac.at