A study of the Empirical Interpolation Method for solving Partial Differential Equations

Abstract

In this thesis the Empirical Interpolation Method (EIM) and the possibility of solving Partial Differential Equations (PDE) using this method are studied. The Empirical Interpolation Method is used to find a polynomial basis, where the basis functions are differentiated analytically and used to find an approximated solution to the Poisson equation through a collocation approach in one and two dimensions. Different boundary conditions and geometries of the domain are tested. The convergence results from solving the Poisson equation with the Empirical Interpolation Method are also compared with Gauss-Lobatto Legendre (GLL) and Finite Difference Method (FDM) in one dimension, and with Gauss-Lobatto Legendre for the square in two dimensions. The convergence rate of the EIM, when used to find an approximated solution of the Poisson equation, was found to be exponential for all geometries. This result is under the condition that the exact solution of the Poisson equation is smooth on the problem domain. The method is found to be competitive with the more traditional Gauss-Lobatto Legendre in one dimension and for the square. The EIM has proven to have multiple advantages compared to GLL in two dimensions. While the GLL only produces a result for a square number of interpolation points, the EIM produces an approximated solution for all number of interpolation points. Also, since the EIM has problem-dependent basis functions and interpolation points, the complexity of the calculations is unaffected by the geometry of the domain. The geometry of the domain is found to be of little importance to the convergence of the method, when assuming a smooth solution of the PDE in question. In this sense, the approach studied here represents an extension of spectral collocation methods to complex domains.