Enforcing Neumann Boundary Conditions with Polynomial Extension Operators to Acheive Optimal Convergence Rates on Polytopial Meshes in the Finite Element Method

2023/01/10 by Cheung, James
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2301.03753

Abstract

In \citecheung2019optimally, the authors presented two finite element methods for approximating second order boundary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear mappings. This was done by enforcing the boundary conditions through judiciously chosen polynomial extension operators. The H¹ error estimates were proven to be optimal for the solutions of both the Dirichlet and Neumann boundary value problems. It was also proven that the Dirichlet problem approximation converges optimally in L². However, optimality of the Neumann approximation in the L² norm was left as an open problem. In this work, we seek to close this problem by presenting new analysis that proves optimal error estimates for the Neumann approximation in the W¹_∞ and L² norms.

Related

• L^∞-error estimates for Neumann boundary value problems on graded meshes
• Optimally accurate higher-order finite element methods on polytopial approximations of domains with smooth boundaries
• Numerical Analysis for Neumann Optimal Control Problems on Convex Polyhedral Domains
• L^∞- and W^1,∞-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain
• Convergence rate of the finite element approximation for extremizers of Sobolev inequalities