Superconvergence in Finite Element Methods and Meshes That are Locally Symmetric with Respect to a Point

1996/04/01 by A. H. Schatz, Ian H. Sloan, I. H. Sloan +2 · 5 citations
Mathematics · Engineering · Computer Science · #Differential Equations and Numerical Methods #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering

paper · doi:10.1137/0733027

Abstract

Consider a second-order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution. Consider also its numerical approximation by standard conforming finite element methods with, for example, fixed degree piecewise polynomials on a quasi-uniform mesh-family (the “h-method”). It will be shown that, if the finite element function spaces are locally symmetric about a point x₀ with respect to the antipodal map x → x₀ - (x - x₀ ), then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x₀ . For piecewise polynomials of even degree, superconvergence occurs in function values; for piecewise polynomials of odd degree, it occurs in derivatives.

Citations

• Interior maximum norm estimates for finite element methods

Cited by

• Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions
• Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations
• Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems
• Novel superconvergence and ultraconvergence structures for the finite volume element method
• L^∞- and W^1,∞-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Related

• Superconvergence of Ritz-Galerkin Finite Element Approximations for Second Order Elliptic Problems
• Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems
• Superconvergence of differential structure for finite element methods on perturbed surface meshes
• Superconvergence of C⁰-Q^k finite element method for elliptic equations with approximated coefficients
• Superconvergence analysis of FEM and SDFEM on graded meshes for a problem with characteristic layers