Using p-Refinement to Increase Boundary Derivative Convergence Rates

TL;DR

This paper introduces a novel finite element method using p-refinement near boundaries to directly improve the convergence rate of boundary derivatives in elliptic problems, enhancing accuracy for physical simulations.

Contribution

It proposes a new p-refinement based finite element approach that increases boundary derivative accuracy without postprocessing, with proven convergence rates in 1D and extended ideas to 2D.

Findings

Convergence rate on p-refined cells depends on the knot connection rate in 1D.

Numerical experiments confirm improved boundary derivative accuracy in 1D and 2D.

Method offers a direct alternative to postprocessing for high-accuracy boundary derivatives.

Abstract

Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating accurate approximations of boundary derivatives of elliptic problems: instead of postprocessing, we describe a new continuous finite element method based on p-refinement of cells adjacent to the boundary to increase the approximation order of the derivative on the boundary itself. We prove that the order of the approximation on the p-refined cells is, in 1D, determined by the rate of convergence at the knot connecting the higher and lower order cells and that this idea can be extended, in some simple settings, to 2D problems. We verify this rate of convergence numerically with a series of experiments in both 1D and 2D.