Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations
The analysis of a finite element method for the numerical solution of singulary perturbed problems which occur in fluid mechanics, gas dynamics and wave propagation is considered. Problems of this form are of mixed hyperbolic-parabolic type. Finite element techniques to examine a difference scheme generated by the Petrov-Galerkin method for advection- diffusion equations in one space variable are used. The scheme is exponentially fitted in the \(x\)-direction. The present paper is devoted to the analysis of a non-lumped method. The authors show that for constant coefficients, when the Courant number is equal to one and the mesh is coarse, the non-lumped scheme is locally higher-order convergent, whereas the lumped scheme is locally only first- order convergent. Under reasonable global assumptions on the behaviour of the exact solution and a certain stability constant, the scheme is shown to be convergent, uniformly in the diffusion parameter, in global energy and \(L^2\) norms and a local discrete \(L^\infty\) norm. These local estimates show that the method does not smear interior and boundary layers excessively and that it is first-order accurate, away from layers, on arbitrary equidistant meshes. Numerical results support these conclusions.
scientific article; zbMATH DE number 699086