An interpolated Galerkin finite element method for the Poisson equation

TL;DR

This paper introduces an interpolated Galerkin finite element method for the Poisson equation that reduces system size and improves condition numbers, achieving optimal convergence and demonstrating advantages over standard methods through numerical tests.

Contribution

The paper develops a new interpolated Galerkin finite element method that decreases unknowns and enhances numerical stability for solving the Poisson equation.

Findings

Reduces the number of unknowns significantly for higher-order elements.

Achieves optimal convergence order in theory and practice.

Demonstrates improved condition numbers and computational efficiency.

Abstract

When solving the Poisson equation by the finite element method, we use one degree of freedom for interpolation by the given Laplacian - the right hand side function in the partial differential equation. The finite element solution is the Galerkin projection in a smaller vector space. The idea is similar to that of interpolating the boundary condition in the standard finite element method. Due to the pointwise interpolation, our method yields a smaller system of equations and a better condition number. The number of unknowns on each element is reduced significantly from $(k^2+3k+2)/2$ to $3k$ for the $P_k$ ($k\ge 3$) finite element. We construct 2D $P_2$ conforming and nonconforming, and $P_k$ ($k\ge3$) conforming interpolated Galerkin finite elements on triangular grids. This interpolated Galerkin finite element method is proved to converge at the optimal order. Numerical tests and comparisons with the standard finite elements are presented, verifying the theory and showing advantages of the interpolated Galerkin finite element method.