A $C^1$-conforming Petrov-Galerkin method for convection-diffusion equations and superconvergence ananlysis over rectangular meshes

TL;DR

This paper introduces a novel $C^1$-conforming Petrov-Galerkin method for convection-diffusion equations, demonstrating optimal error estimates and superconvergence properties on rectangular meshes through rigorous analysis and numerical validation.

Contribution

A new $C^1$-conforming Petrov-Galerkin method with superconvergence analysis for convection-diffusion equations on rectangular meshes.

Findings

Optimal error estimates in $L^2$, $H^1$, $H^2$ norms.

Superconvergence at mesh nodes, Jacobi polynomial roots, Lobatto and Gauss lines.

Numerical experiments confirm theoretical results.

Abstract

In this paper, a new $C^1$-conforming Petrov-Galerkin method for convection-diffusion equations is designed and analyzed. The trail space of the proposed method is a $C^1$-conforming ${\mathbb Q}_k$ (i.e., tensor product of polynomials of degree at most $k$) finite element space while the test space is taken as the $L^2$ (discontinuous) piecewise ${\mathbb Q}_{k-2}$ polynomial space. Existence and uniqueness of the numerical solution is proved and optimal error estimates in all $L^2, H^1, H^2$-norms are established. In addition, superconvergence properties of the new method are investigated and superconvergence points/lines are identified at mesh nodes (with order $2k-2$ for both function value and derivatives), at roots of a special Jacobi polynomial, and at the Lobatto lines and Gauss lines with rigorous theoretical analysis. In order to reduce the global regularity requirement, interior a priori error estimates in the $L^2, H^1, H^2$-norms are derived. Numerical experiments are presented to confirm theoretical findings.