Numerical analysis of Darcy problem on surfaces

Politecnico di Milano, Department of Mathematics. Piazza Leonardo da Vinci 32, 20133 Milan, Italy.
alberto.ferroni@polimi.it; luca.formaggia@polimi.it; alessio.fumagalli@polimi.it

Received: 11 March 2015
Revised: 19 November 2015
Accepted: 17 December 2015

Abstract

Surface problems play a key role in several theoretical and applied fields. In this work the main focus is the presentation of a detailed analysis of the approximation of the classical porous media flow problem: the Darcy equation, where the domain is a regular surface. The formulation considers the mixed form and the numerical approximation adopts a classical pair of finite element spaces: piecewise constant for the scalar fields and Raviart–Thomas for vector fields, both written on the tangential space of the surface. The main result is the proof of the order of convergence where the discretization error, due to the finite element approximation, is coupled with a geometrical error. The latter takes into account the approximation of the real surface with a discretized one. Several examples are presented to show the correctness of the analysis, including surfaces with boundary.