Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods

TL;DR

This paper introduces a method to enforce physical property constraints like positivity and monotonicity in finite element solutions of PDEs, ensuring valid and physically meaningful results without sacrificing convergence rates.

Contribution

It develops a convex optimization-based postprocessing filter that preserves structural properties in spectral/hp element methods for PDEs, applicable to continuous and discontinuous Galerkin schemes.

Findings

Constraints can be enforced without affecting convergence rates.

The method effectively preserves positivity and monotonicity in test PDEs.

The approach acts as a norm-decreasing filter, ensuring physical validity.

Abstract

Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the method's efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.