A first order system least squares method for the Helmholtz equation

Abstract

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always leads to a Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of the standard finite element methods, we give an error analysis to the hp-version of the FOSLS method where the dependence on the mesh size h, the approximation order p, and the wave number k is given explicitly. In particular, under some assumption of the boundary of the domain, the L 2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p is sufficiently small and the polynomial degree p is at least O(log k). Numerical experiments are given to verify the theoretical results.2000 Mathematics Subject Classification. 65N30, 65L12.

Analysis of the hp-Version of a First Order System Least Squares Method for the Helmholtz Equation

“…We mimic the procedure of [20, Thm. 5.5] and [4,Lemma 4.7]. First consider for each element K ∈ T h the constant C K given by…”

“…In particular, as discussed in more detail in [21,6], the analyses [11,12,13,1,20,21,6] show that high order methods are much better suited for the high-frequency case of large k than low order Maximilian Bernkopf Technische Universität Wien, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, A-1040 Vienna, e-mail: maximilian.bernkopf@tuwien.ac.at Jens Markus Melenk Technische Universität Wien, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, A-1040 Vienna, e-mail: melenk@tuwien.ac.at methods. Alternatives to the classical Galerkin methods that are still based on high order methods include stabilized methods for Helmholtz [8,9,10,28], hybridizable methods [3], least-squares type methods [4,15] and Discontinuous Petrov Galerkin methods, [24,5]. An attractive feature of least squares type methods is that the resulting linear system is always solvable and that they feature quasi-optimality, albeit in some nonstandard residual norms.…”

“…In the present paper, we show for the least squares method (4) an a priori estimate in the more tractable L 2 (Ω )-norm under the scale resolution condition (35). For that, we closely follow [4]. Our key refinement over [4] is an improved regularity estimate for the solution of a suitable dual problem (cf.…”

“…For that, we closely follow [4]. Our key refinement over [4] is an improved regularity estimate for the solution of a suitable dual problem (cf. Lemma 3.1 vs. [4,Lemma 5.1]) that allows us to establish the improved p-dependence in the L 2 (Ω )-error estimate (cf.…”

“…Lemma 3.1 vs. [4,Lemma 5.1]) that allows us to establish the improved p-dependence in the L 2 (Ω )-error estimate (cf. Theorem 5.1 vs. [4,Thm. 2.5]).…”