What type of study is this?

This is a Quantitative Study study.

September 30, 2025

hp-error analysis of a continuous Petrov-Galerkin time-stepping scheme for linear parabolic equations

Key Points

Achieves numerical convergence with robust error estimates in L2(L2)- and L∞(L2)-norms for parabolic equations.
Utilizes continuous Petrov-Galerkin scheme alongside finite element methods to enhance time-stepping accuracy.
Exponential temporal convergence is achieved via geometrically graded time meshes and higher polynomial degrees.
Validation through numerical experiments corroborates theoretical convergence rates, enhancing practical applicability.

Abstract

We present a fully discrete hp-version numerical method for second-order linear parabolic equations, combining a continuous Petrov-Galerkin (CPG) time-stepping scheme with a standard continuous Galerkin (CG) finite element method in space. Our analysis provides a priori error estimates in the L2(L2)- and L∞(L2)-norms, where all constants are fully robust—independent of temporal and spatial discretization parameters. For solutions with initial singularities at t = 0, exponential temporal convergence is achieved through geometrically graded time meshes and linearly increasing polynomial degrees. Numerical experiments confirm the theoretical convergence rates.

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