CONNECTIONS BETWEEN FINITE DIFFERENCE AND FINITE ELEMENT APPROXIMATIONS FOR A CONVECTION-DIFFUSION PROBLEM.

BACUTA, CRISTINA; BACUTA, CONSTANTIN

We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. 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. This new approach allows for improving well-known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.

FINITE difference method; FINITE differences; DISCRETIZATION methods; FINITE element method; LINEAR systems

Romanian Journal of Pure & Applied Mathematics / Revue Roumaine de Mathematiques Pures et Appliquees, 2024, Vol 69, Issue 3/4, p353

0035-3965

Academic Journal

10.59277/RRMPA.2024.353.374