[Paper Review] Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

Pratyuksh Bansal, Andrea Moiola+2 more|arXiv (Cornell University)|Feb 26, 2020

Advanced Numerical Methods in Computational Mathematics45 references24 citations

TL;DR

This paper presents a space–time discontinuous Galerkin (DG) method for the linear acoustic wave equation in 2D polygonal domains with point singularities due to corners or multi-material interfaces. It establishes optimal convergence rates using local spatial mesh grading and introduces a novel sparse tensor space–time DG scheme that achieves optimal convergence with work scaling like one elliptic solve on the finest grid, even in the presence of singularities.

ABSTRACT

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\Omega$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.

Motivation & Objective

• Address the challenge of optimal convergence in wave equations with point singularities in polygonal domains.
• Develop a space–time DG formulation stable under arbitrary time and space step sizes, even with local mesh refinement.
• Design a sparse tensor space–time DG scheme that maintains optimal convergence while reducing degrees of freedom.
• Establish convergence rate bounds in corner-weighted Sobolev spaces for solutions with conical singularities.
• Demonstrate numerically optimal convergence for both full and sparse space–time DG schemes in singular and smooth regimes.

Proposed method

• Adopts an unconditionally stable space–time DG variational formulation from prior work [32] for the second-order wave equation.
• Applies local isotropic mesh refinement towards spatial corners and multi-material interfaces to resolve point singularities.
• Uses corner-weighted Sobolev spaces of Kondrat’ev type to model solution regularity in non-smooth domains.
• Proposes a sparse tensor space–time DG scheme based on the combination formula applied to multiple DG solutions with varying time and space steps.
• Employs anisotropic space–time DG discretizations with graded meshes to balance spatial and temporal resolution near singularities.
• Leverages unconditional stability to allow CFL-violating combinations of time and space steps in the combination formula.

Experimental results

Research questions

• RQ1 Can optimal convergence rates be achieved for space–time DG schemes in the presence of point singularities in 2D polygonal domains?
• RQ2 Does local spatial mesh grading towards corners or interfaces restore optimal convergence despite reduced solution regularity?
• RQ3 Can a sparse tensor space–time DG scheme achieve optimal convergence with significantly reduced degrees of freedom compared to the full tensor version?
• RQ4 How does the combination formula perform when applied to space–time DG schemes with varying time and space steps, especially in the presence of singularities?
• RQ5 What is the asymptotic work complexity of the sparse space–time DG scheme relative to solving a single elliptic problem on the finest grid?

Key findings

• Optimal convergence rates of order $ O(h^p) $ are proven for the full tensor space–time DG scheme with local isotropic mesh grading towards corners.
• The sparse tensor space–time DG scheme achieves optimal convergence order $ O(M_L^{-p}) $, where $ M_L $ is the number of degrees of freedom, with work scaling like one elliptic solve on the finest spatial grid.
• Numerical experiments confirm optimal convergence rates for both smooth and singular solutions, with the sparse scheme outperforming the full tensor version in error-to-dof ratio below 1% relative error.
• The combination formula enables optimal convergence even with CFL-violating combinations of time and space steps, made possible by the unconditional stability of the underlying DG scheme.
• For polynomial degree $ p = 2 $, the sparse scheme achieves convergence rate $ O(M_L^{-1.5}) $, while the full tensor scheme achieves $ O(M_L^{-1}) $, indicating superior efficiency.
• The method extends to piecewise-homogeneous media and allows exact evaluation of cell contributions when coefficients are piecewise constant.