Robust Adaptive $hp$ Discontinuous Galerkin Finite Element Methods for the Helmholtz Equation

Abstract

This paper presents an hp a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree p and the wave number k. For the discretization, we consider a discontinuous Galerkin formulation that is unconditionally well posed. The a posteriori error analysis is based on the technique of equilibrated fluxes applied to a shifted Poisson problem, with the error due to the nonconformity of the discretization controlled by a potential reconstruction. We prove that the error estimator is both reliable and efficient, under the condition that the initial mesh size and polynomial degree is chosen such that the discontinuous Galerkin formulation converges, i.e., it is out of the regime of pollution. We confirm the efficiency of an hp-adaptive refinement strategy based on the presented robust a posteriori error estimator via several numerical examples.

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

“…appears for Γ A = ∅. Importantly, in contrast to (1.3b), the constant C(κ T K ) in (1.5b) is independent of the polynomial degree p (only depends on the local shape-regularity parameter κ T K ), and, in contrast to [20], the scaled L 2 (Ω) and L 2 (Γ A ) terms are included. Since C up is also independent of p, we conclude that our a posteriori error estimator is p-robust in all regimes.…”

“…The typical dependence (1.2) encourages the use of finite elements with high polynomial degree p to solve problems with high wavenumber k, and this is a usual practice, cf. [2,6,14,20,44,54] and the references therein.…”

“…In this case, an unknown-constant-free upper bound is obtained for elliptic problems. Moreover, it has been recently shown [7,29,30] that such a strategy provides a p-robust estimator which means that the lower bound does not depend on the polynomial degree p. The recent work [20] proposes to use this type of estimators for discontinuous Galerkin discretizations of the Helmhotz problem. The authors consider a simpler configuration of (1.1) with Γ D = ∅ and d = 2 and evaluate the error in the H 1 (Ω)-seminorm |u − u h | 1,Ω only.…”

“…The factor √ 2 can be further decreased to the optimal value of 1 modulo additional technical developments that we detail in Theorem 2.3 below. Thus, asymptotically, (1.5a) is unknownconstant-free, in improvement of (1.3a), and additionally the entire energy norm (including the scaled L 2 (Ω) and L 2 (Γ A ) terms) is controlled in contrast to [20]. Though the asymptotic nature of the approximability factor σ ba is known, cf.…”

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

“…2 appears for Γ A = ∅. Importantly, in contrast to (1.3b), the constant C(κ T K ) in (1.5b) is independent of the polynomial degree p (only depends on the local shape-regularity parameter κ T K ), and, in contrast to [20], the scaled L 2 (Ω) and L 2 (Γ A ) terms are included. Since C up is also independent of p, we conclude that our a posteriori error estimator is p-robust in all regimes.…”