Should multilevel methods for discontinuous Galerkin discretizations use discontinuous interpolation operators?

TL;DR

This paper investigates whether using discontinuous interpolation operators in multilevel preconditioners for DG discretizations improves solver efficiency, optimizing the operator to achieve faster convergence and better spectral properties.

Contribution

It introduces an optimal discontinuous interpolation operator with a tunable parameter for DG methods, enhancing multilevel solver performance and spectral clustering.

Findings

Optimal discontinuous interpolation improves solver speed.

Spectral clustering with high multiplicity is achieved.

Method extends effectively to higher dimensions.

Abstract

Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such solvers is the interpolation operator to transfer information from the coarse to the fine grid. Classical interpolation operators give continuous interpolated values, but since DG solutions are naturally discontinuous, one might wonder if one should not use discontinuous interpolation operators for DG discretizations. We consider a discontinuous interpolation operator with a parameter that controls the discontinuity, and determine the optimal choice for the discontinuity, leading to the fastest solver for a specific 1D symmetric interior penalty DG discretization model problem. We show in addition that our optimization delivers a perfectly clustered spectrum with a high geometric multiplicity, which is very advantageous for a Krylov solver using the method as its preconditioner. Finally, we show the applicability of the optimal choice to higher dimensions.