[Paper Review] Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems

Yuanyuan Zhang, Zhongying Chen|arXiv (Cornell University)|Oct 28, 2016

Advanced Numerical Methods in Computational Mathematics2 references3 citations

TL;DR

This paper develops a unified convergence theory for nonconforming finite volume methods (FVMs) using nonconforming finite element spaces as trial functions for second-order elliptic boundary value problems. It establishes optimal error estimates in mesh-dependent $H^1$-norm and $L^2$-norm for two specific schemes—Crouzeix-Raviart and hybrid Wilson FVM—under regular mesh conditions, with convergence orders of $O(h)$ and $O(h^2)$, respectively.

ABSTRACT

This paper is devoted to analyze of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent $H^1$-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.

Motivation & Objective

• To establish a general convergence framework for nonconforming finite volume methods applied to second-order elliptic problems.
• To address the challenges in analyzing nonconforming FVMs, including uniform boundedness, ellipticity, and nonconforming consistency error.
• To analyze two specific nonconforming FVMs: Crouzeix-Raviart and hybrid Wilson FVM, under a unified Petrov-Galerkin formulation.
• To derive optimal error estimates in both mesh-dependent $H^1$-norm and $L^2$-norm for these schemes.
• To validate theoretical findings with numerical experiments on Poisson problems with exact solutions.

Proposed method

• Formulates nonconforming FVMs as Petrov-Galerkin methods with nonconforming finite element trial spaces and piecewise constant test functions over dual partitions.
• Derives a general convergence theorem based on discrete norm inequalities, uniform boundedness, and uniform ellipticity of the discrete bilinear forms.
• Applies the framework to the Crouzeix-Raviart FVM, proving that the nonconforming error vanishes, leading to optimal $O(h)$ convergence in the mesh-dependent $H^1$-norm.
• Analyzes the hybrid Wilson FVM using a test space combining piecewise constants and linearly independent functions from the trial space, enabling $L^2$-norm error analysis.
• Employs Green’s formula and Cauchy-Schwarz inequality to bound residual terms and establish $L^2$-error bounds.
• Uses mesh regularity (minimum angle condition) to control approximation errors and ensure optimal convergence rates.

Experimental results

Research questions

• RQ1 Can a unified theoretical framework be developed for analyzing nonconforming finite volume methods for second-order elliptic problems?
• RQ2 What conditions ensure uniform boundedness and uniform ellipticity of the discrete bilinear forms in nonconforming FVMs?
• RQ3 How can the nonconforming consistency error be controlled or eliminated in the analysis?
• RQ4 What is the optimal convergence rate of the Crouzeix-Raviart FVM in the mesh-dependent $H^1$-norm under regular mesh assumptions?
• RQ5 Does the hybrid Wilson FVM achieve the same $L^2$-convergence order as the corresponding Wilson finite element method?

Key findings

• The nonconforming Crouzeix-Raviart FVM achieves optimal $O(h)$ convergence in the mesh-dependent $H^1$-norm, with the nonconforming error term identically zero.
• The hybrid Wilson FVM exhibits $O(h)$ convergence in the mesh-dependent $H^1$-norm and $O(h^2)$ convergence in the $L^2$-norm, matching the convergence rates of the corresponding Wilson finite element method.
• Numerical experiments confirm the theoretical $O(h)$ convergence order in the $H^1$-norm for the Crouzeix-Raviart FVM across various triangulations with minimum angles $45^ ext{circ}$, $\approx18.43^\circ$, and $\approx2.86^\circ$.
• The $L^2$-error estimate for the hybrid Wilson FVM is derived via techniques adapted from lower-order FVM and Wilson FEM analyses, achieving $O(h^2)$ convergence under regular mesh conditions.
• The convergence analysis relies on discrete norm inequalities and bounds on the reaction term and gradient terms using Green’s formula and Cauchy-Schwarz inequality.
• The theoretical results are validated numerically using a Poisson problem on $[0,1] \times [0,1]$ with exact solution $u(x,y) = -x(x-1)y(y-1)$, showing consistent convergence orders across refined meshes.