$C^0$-hybrid high-order methods for biharmonic problems

TL;DR

This paper introduces and analyzes $C^0$-conforming hybrid high-order methods for biharmonic problems, achieving high accuracy with reduced regularity requirements and demonstrating efficiency through numerical comparisons.

Contribution

The paper develops novel $C^0$-conforming HHO methods for biharmonic problems with improved error analysis and lower regularity assumptions, extending applicability beyond existing methods.

Findings

Achieves $O(h^{k+1})$ $H^2$-error estimates for smooth solutions.

Reduces minimal regularity requirements for error analysis.

Numerical results confirm the efficiency of the proposed methods.

Abstract

We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns which are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns which are polynomials of order $k\ge0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has broader applicability than just $C^0$-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known $C^0$-conforming interior penalty discontinuous Galerkin (IPDG) methods as well. The present technique does not require bubble functions or a $C^1$-smoother to evaluate the right-hand side in case of rough loads. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.