Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation

Abstract

We propose and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides additional control over the time derivative of the discrete solution. We derive $hp$-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. Numerical experiments validate our theoretical results.

References in corpus (10)

• Space-Time Isogeometric Analysis of Parabolic Evolution Equations
• A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation
• A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems
• Discontinuous Galerkin Methods with Trefftz Approximation
• Improved rates for a space-time FOSLS of parabolic PDEs
• Space-time virtual elements for the heat equation
• Design and performance of a space-time virtual element method for the heat equation on prismatic meshes
• On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation
• A space-time DG method for the Schrödinger equation with variable potential
• Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems