Trivariate Interpolated Galerkin Finite Elements for the Poisson Equation

When applying finite element method to the Poisson equation on a domain in \(\mathbb {R}^3\), we replace some Lagrange nodal basis functions by bubble functions whose dual functionals are the values of the Laplacian. To compute the coefficients of these Laplacian basis functions instead of solving a large linear system, we interpolate the right hand side function in the Poisson equation. The finite element solution is then the Galerkin projection on a smaller vector space. We construct a qudratic and a cubic nonconforming interpolated finite elements, and quartic and higher degree conforming interpolated finite elements on arbitrary tetrahedral partitions. The main advantage of our method is that the number of unknowns that require solving a large system of equations on each element is reduced. We show that the interpolated Galerkin finite element method retains the optimal order of convergence. Numerical results confirming the theory are provided as well as comparisons with the standard finite elements.