Efficient numerical approach for solving the diffusion equation with variable coefficients

Abstract

We propose and investigate an N-layered diffusion model with piecewise constant coefficients for approximating the exact solution of the diffusion equation with depth-dependent material coefficients in the spatial frequency domain. It is shown that this numerical approach, which is quite easy in view of the implementation, exhibits a convergence rate of O ( N − 2 ) in the discrete L ∞ -norm and hence provides an interesting alternative to frequently used numerical approaches such as finite element or finite difference methods. For comparison purposes, we take into account the classical finite element approach under the use of continuous linear basis functions as well as a recently reported nonconforming finite element discretization - the weak Gakerkin method - that employs discontinuous functions in form of locally defined polynomials.