[Paper Review] Superconvergence of high order finite difference schemes based on variational formulation for elliptic equations

Hao Li, Xiangxiong Zhang|arXiv (Cornell University)|Apr 2, 2019

Advanced Numerical Methods in Computational Mathematics12 references4 citations

TL;DR

This paper establishes superconvergence of high-order finite difference schemes derived from the $C^0$-$Q^k$ finite element method using $(k+1) imes(k+1)$ Gauss-Lobatto quadrature. By replacing integrals in the variational formulation with this quadrature, the scheme achieves $(k+2)$-th order accuracy in the discrete $L^2$-norm for elliptic equations with Dirichlet boundary conditions, demonstrating that function values at Gauss-Lobatto points converge at superoptimal rates.

ABSTRACT

The classical continuous finite element method with Lagrangian $Q^k$ basis reduces to a finite difference scheme when all the integrals are replaced by the $(k+1) imes (k+1)$ Gauss-Lobatto quadrature. We prove that this finite difference scheme is $(k+2)$-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values.

Motivation & Objective

• To establish superconvergence of function values at Gauss-Lobatto points for high-order finite difference schemes derived from the $C^0$-$Q^k$ finite element method.
• To demonstrate that using $(k+1) imes(k+1)$ Gauss-Lobatto quadrature in the variational formulation yields a finite difference scheme with $(k+2)$-th order accuracy in the discrete $L^2$-norm.
• To provide a practical and efficient finite difference implementation that preserves superconvergence without sacrificing accuracy or symmetry.
• To extend the analysis to variable coefficient problems, convection terms, and three-dimensional problems, confirming the same convergence rates.

Proposed method

• Derive a finite difference scheme by replacing all integrals in the $C^0$-$Q^k$ finite element formulation with $(k+1) imes(k+1)$ Gauss-Lobatto quadrature, which matches the degrees of freedom of the $Q^k$ basis.
• Formulate the discrete problem as $A_h(u_h, v_h) = raket{f, v_h}_h$ for all $v_h o V_0^h$, where $A_h$ and $raket{f, v_h}_h$ use Gauss-Lobatto quadrature for the bilinear and linear forms.
• Prove that the resulting scheme is $(k+2)$-th order accurate in the discrete $L^2$-norm under smoothness assumptions on the solution and coefficients.
• Use variational analysis and discrete Green’s function techniques to bound the error at Gauss-Lobatto points, leveraging the superconvergence properties of the quadrature nodes.
• Verify the theoretical results numerically in 2D and 3D for Poisson equations, variable coefficient problems, and convection-diffusion equations with Dirichlet and Neumann conditions.
• Implement the scheme efficiently using matrix assembly via tensor-product quadrature and solve the linear system via iterative or direct solvers, as demonstrated in Section 7.4.

Experimental results

Research questions

• RQ1 Can a high-order finite difference scheme derived from the $C^0$-$Q^k$ finite element method via Gauss-Lobatto quadrature achieve superconvergence for function values at the quadrature points?
• RQ2 Does the use of $(k+1) imes(k+1)$ Gauss-Lobatto quadrature preserve the $(k+2)$-th order accuracy in the discrete $L^2$-norm for elliptic problems with Dirichlet boundary conditions?
• RQ3 How does the scheme perform for variable coefficient problems, convection terms, and in three dimensions, and does the superconvergence rate persist?
• RQ4 Can the resulting scheme be efficiently implemented as a standard finite difference stencil while maintaining high-order accuracy and symmetry?
• RQ5 What is the maximum norm superconvergence behavior at Gauss-Lobatto points, and how does it compare to the $L^2$-norm result?

Key findings

• The finite difference scheme derived from $C^0$-$Q^k$ FEM using $(k+1) imes(k+1)$ Gauss-Lobatto quadrature achieves $(k+2)$-th order accuracy in the discrete $L^2$-norm for elliptic equations with Dirichlet boundary conditions.
• For $k=2$, the scheme is fourth-order accurate in both $L^2$ and $L^rown$ norms, as confirmed by numerical experiments on 2D and 3D Poisson problems.
• The scheme maintains symmetry of the stiffness matrix and allows for efficient matrix assembly, making it practical for high-order simulations.
• Numerical results show convergence orders of approximately 3.95–3.96 in $L^2$ and 3.92–3.94 in $L^rown$ for $k=2$, confirming the theoretical $(k+2)$-th order superconvergence.
• The superconvergence result extends to three-dimensional problems and variable coefficient problems, with observed convergence orders of 4.00–4.04 in $L^2$ and 4.00 in $L^rown$ for $k=2$.
• The scheme remains effective for convection-diffusion problems with incompressible velocity fields, achieving $O(h^{k+2})$ convergence in both norms.