Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems

Abstract: The $h$-version of the finite-element method ($h$-FEM) applied to the high-frequency Helmholtz equation has been a classic topic in numerical analysis since the 1990s. It is now rigorously understood that (using piecewise polynomials of degree $p$ on a mesh of a maximal width $h$) the conditions "$(hk)^p \rho$ sufficiently small" and "$(hk)^{2p} \rho$ sufficiently small" guarantee, respectively, $k$-uniform quasioptimality (QO) and bounded relative error (BRE), where $\rho$ is the norm of the solution operator with $\rho\sim k$ for non-trapping problems. Empirically, these conditions are observed to be optimal in the context of $h$-FEM with a uniform mesh. This paper demonstrates that QO and BRE can be achieved using certain non-uniform meshes that violate the conditions above on $h$ and involve coarser meshes away from trapping and in the perfectly matched layer (PML). The main theorem details how varying the meshwidth in one region affects errors both in that region and elsewhere. One notable consequence is that, for any scattering problem (trapping or nontrapping), in the PML one only needs $hk$ to be sufficiently small; i.e. there is no pollution in the PML. The motivating idea for the analysis is that the Helmholtz data-to-solution map behaves differently depending on the locations of both the measurement and data, in particular, on the properties of billiards trajectories (i.e. rays) through these sets. Because of this, it is natural that the approximation requirements for finite-element spaces in a subset should depend on the properties of billiard rays through that set. Inserting this behaviour into the latest duality arguments for the FEM applied to the high-frequency Helmholtz equation allows us to retain detailed information about the influence of $\textit{both}$ the mesh structure $\textit{and}$ the behaviour of the true solution on local errors in FEM.

Essay: Non-uniform Finite-Element Meshes Defined by Ray Dynamics for Helmholtz Problems

This paper addresses the challenge of achieving quasioptimality and bounded relative error in finite-element methods (FEM) applied to high-frequency Helmholtz problems, through the use of non-uniform meshes. The keen focus is on how the mesh's non-uniformity can be strategically designed by incorporating insights from ray dynamics, thus allowing advances in FEM's performance beyond traditional uniform mesh approaches.

High-Frequency Helmholtz Equations and FEM

The study of the $h$-version of the finite-element method ($h$-FEM), particularly in the context of high-frequency Helmholtz equations, is well-established in numerical analysis. Typically, for FEM, conditions such as $(hk)^p \rho$ being sufficiently small are often necessary to ensure $k$-uniform quasioptimality (QO), while $(hk)^{2p} \rho$ sufficiently small is considered requisite for bounded relative error (BRE). Here, $hk$ relates to mesh size and $k$ to frequency, with $\rho$ reflecting the norm of the solution operator. Past research generally assumes that such criteria are only met with uniform mesh distributions.

Non-Uniform Mesh Innovation

This paper seeks to transcend the limitations inherent to uniform meshes by introducing a non-uniform mesh strategy that adapts according to the dynamics of ray trajectories (billiard paths) in different regions of the domain. In pragmatic terms, the novelty lies in demonstrating that certain non-uniform meshes can violate the standard conditions on mesh size yet still achieve QO and BRE.

Theoretical Insights and Methodology

Key to the methodology is a differentiation between regions based on ray dynamics—identifying areas of trapping, visibility, and invisibility related to billiard trajectories, and adjusting mesh density accordingly. The theoretical insights apply advanced duality arguments to finite-element spaces, allowing a nuanced understanding of how varying mesh size impacts error distribution and propagation.

Relevantly, the paper establishes that within the perfectly matched layer (PML), even for scattering problems that involve trapping, QO and BRE can be maintained with a less strict mesh criterion. This realization removes the pollution effect, altering previously held constraints that required smaller mesh sizes with increasing $k$.

Results and Implications

Numerically and theoretically, the work confirms that meshes extremely coarser than traditionally considered sufficient can nonetheless provide the desired finite-element solution accuracy. This outcome contravenes and thus prompts a reevaluation of existing empirical mesh size recommendations. The research not only marks a critical shift in $h$-FEM for Helmholtz problems but also establishes a groundwork for exploiting domain-specific dynamics to refine numerical solutions.

The implications stretch across computational techniques in wave propagation problems, suggesting significant computational savings and accuracy enhancements in domains varying from acoustic to electromagnetic wave analysis. Specifically, it emphasizes the importance of integrating dynamic cues into mesh design, with potential to generalize these insights to other domains governed by partial differential equations.

Conclusion

By applying sophisticated mathematical modeling and leveraging physics-based domain properties, the paper makes a substantial contribution to the field of numerical methods for wave-based problems. The extension of FEM beyond uniform mass constraints unveils new avenues for optimizing computational resources and improving solution accuracy, heralding both practical and theoretical advancements in engineering and applied mathematics. Future studies could further explore other dynamic properties intrinsic to complex domains, enhancing mesh strategies across different mathematical contexts.