The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains

Abstract

We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

Figures (4)

• Figure 2.1: Numerical example: sound-soft scattering with $p=2$
• Figure 2.5: Numerical example: penetrable obstacle with $p=2$
• Figure 4.1: Example of a curved domain $\Omega$ with two subdomains $Q_\pm$. The mesh $\mathcal{T}_h$ consisting of straight elements induces an approximate domain $\Omega^h$ partitioned into $Q_\pm^h$. Here, for each $K \in \mathcal{T}_h$, $\mathcal{F}_K: \widehat{K} \to K$ is the affine map mapping the reference element into $K$. The mapping $\Psi^h$ maps $\Omega^h$ onto $\Omega$, and, in particular, maps each element $K_j$ onto $\widetilde{K}_j$. In this particular example, $\Psi^h|_{K_j}$ is the identity map for $j=2,4,6$ and $8$, and is a non-affine mapping on the remaining elements. $\Phi^h$ correspondingly maps $\widetilde{K}_j$ onto $K_j$.
• Figure 4.2: Example illustrating Assumption \ref{['assumption_straight_fem']}. The mesh $\mathcal{T}_h$ is depicted on the left. A mapping $\Psi^h$ satisfying Assumption \ref{['assumption_straight_fem']} is represented in the middle. A mapping $\Psi^h$ not satisfying Assumption \ref{['assumption_straight_fem']} is represented on the right.

Theorems & Definitions (34)

• definition 1: Sound-soft scattering by a 2-d ball
• definition 2: Scattering by a penetrable 2-d ball
• definition 3: PML approximation to sound-soft scattering by a 2-d ball
• definition 4: PML approximation to scattering by a penetrable 2-d ball
• remark 1: The relationship between \ref{['eq_Garding']} and the standard Gå rding inequality
• theorem 1: Abstract elliptic-projection-type argument with variational crime
• lemma 1
• lemma 2
• proof : Proof of Theorem \ref{['thm_abs1']} using Lemmas \ref{['lem:abs1']} and \ref{['lem:abs2']}
• proof : Proof of Lemma \ref{['lem:abs2']}