Exponential finite elements for diffusion-advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non-isoparametric exponentially-varying interpolation functions, based on exact, one- and two-dimensional solutions of the Laplace-transformed differential equation. Two eight-noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight- and 12-noded polynomial elements. The exponential element based on two-dimensional analytical solutions fails basic tests of convergence. The one based on one-dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight-noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions