Connections Between Finite Difference and Finite Element Approximations

TL;DR

This paper explores the mathematical connections between finite difference and finite element methods for boundary value problems, revealing how their solutions relate and providing error estimates and extensions to higher dimensions.

Contribution

It establishes a link between finite difference and finite element approximations, including a new formula for the inverse stiffness matrix on non-uniform meshes and error analysis.

Findings

Finite element interpolant coincides with finite element approximation in 1D.

Derived a formula for the inverse stiffness matrix on non-uniform meshes.

Provided error estimates for the difference between the two methods.

Abstract

We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods, and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.