An adaptive mesh refinement strategy to ensure quasi-optimality of finite element methods for self-adjoint Helmholtz problems

TL;DR

This work tackles uniform quasi-optimality for Galerkin discretizations of the self-adjoint Helmholtz problem by combining a $T$-coercivity framework with an eigenvalue-aware adaptive mesh refinement strategy. It analyzes both $H^1$-conforming and Crouzeix–Raviart discretizations, showing stability and quasi-optimality when the discrete eigenvalues straddle the wave-number via $\lambda_h^{(i_*)} < k^2 < \lambda_h^{(i_* olinebreak[4] + 1)}$, and provides explicit mesh-size conditions tied to eigenvalue approximation errors. A practical algorithm is proposed to generate quasi-optimal meshes using a residual-based estimator, with CR benefiting from guaranteed eigenvalue bounds to certify the index $i_*$. Extensive numerical experiments on canonical geometries demonstrate effective adaptive refinement and quasi-optimal performance with minimal degrees of freedom, with implementations in Python using Firedrake and ngsPETSc.

Abstract

It is well known that the quasi-optimality of the Galerkin finite element method for the Helmholtz equation is dependent on the mesh size and the wave-number. In the literature, different criteria have been proposed to ensure uniform quasi-optimality of the discretisation. In the present work, we study the uniform quasi-optimality of $H^1$ conforming and non-conforming Crouzeix-Raviart discretisation of the self-adjoint Helmholtz problem. In particular, we propose an adaptive scheme, coupled with a residual-based indicator, for generating guaranteed quasi-optimal meshes with minimal degrees of freedom.

Figures (7)

• Figure 1: We assume the discrete eigenvalues to be ordered but, depending on the discretisation, it might be the case that $\lambda_h^{(i_{\ast})} > k^2$ (left) or $\lambda^{(i_{\ast}+1)} < k^2$ (right). We achieve T$_h$-coercivity by choosing $h$ small enough such that $\lambda^{(i_{\ast})}_h < k^2$ and $\lambda^{(i_{\ast}+1)}_h > k^2$, respectively.
• Figure 2: $L^2$-error of the approximation of the Helmholtz problem with homogeneous Dirichlet boundary conditions against a computed reference solution, computed with higher polynomial degree, for $k^2 \in \{100,144,225,400\}$. The vertical lines indicate when $\lambda_h^{(i_\ast)} < k^2$ after which we expect quasi-optimality.
• Figure 3: We compare the approximation error using Crouzeix-Raviart elements and conforming $\mathcal{P}^1$-elements. For increasing wave number $k^2$, we compute the $L^2$-error against a reference solution computed with a higher order $H^1$-conforming method. The dashed lines indicate the mesh size at which the criterion $\lambda_h^{(i_{\ast})} < k^2$ (for the $\mathcal{P}^1$-case) or $\lambda_h^{(i_{\ast}+1)} > k^2$ (for the Crouzeix-Raviart case). For the first two wave-numbers (top row) the relevant eigenvalues are $\lambda^{(i_{\ast})} = 197.39$ and $\lambda^{(i_{\ast}+1)} = 246.74$, whereas for the other wave-numbers (bottom row) the relevant eigenvalues are $\lambda^{(i_{\ast})} = 1224.09$ and $\lambda^{(i_{\ast}+1)} = 1233.70$.
• Figure 4: On the left, we compare the approximation of the eigenvalues $\lambda^{(i)} = 197.39$ (solid) and $\lambda^{(i+1)} = 246.74$ (dashed) with Crouzeix-Raviart and $\mathcal{P}^1$-elements. For $k^2 \approx 239.63$, cf. Figure \ref{['fig:CRvsP1']}, the $\mathcal{P}^1$-discretisation reaches quasi-optimality (dashed circle) before the Crouzeix-Raviart discretisation. On the right, we compare mesh size required for the criterion $\lambda_h^{(i_{\ast})} < k^2 < \lambda_h^{(i_{\ast}+1)}$ to be met with the respective approximations for wave numbers $k^2\in [100,500]$. When the reference eigenvalue $\lambda^{(i_{\ast})}$ changes, the requirements on the mesh size become suddenly stricter for the $\mathcal{P}^1$-discretisation and more relaxed for the Crouzeix-Raviart discretisation, consider for example the moment the wave number becomes larger than the eigenvalue $\lambda = 246.74$.
• Figure 5: We solve the Helmholtz problem on a tuning fork domain using a sequence of uniformly refined meshes, with $k^2$ equal to $100$ and data a Gaussian bump $f(x,y) = 5\times 10^4 \exp \left[{-(40^2)\cdot ((x-10)^2+(y-\frac{7}{4})^2)}\right]$. The criterion here proposed suggest that mesh $(c)$ is the first one to guarantee quasi-optimality.

Theorems & Definitions (17)

• Definition 1: T-coercivity
• Definition 2
• Theorem 3: Thm. 2 of Cia12
• Lemma 4
• proof
• Theorem 5
• proof
• Corollary 6
• proof
• Remark 7