Application of an element-by-element BiCGSTAB iterative solver to a monotonic finite element model

This paper deals with the advection-diffusion equation in adaptive meshes. The main feature of the present finite element model is the use of Legendre-polynomials to span finite element spaces. The success that this model gives good resolutions to solutions in regions of boundary and interior layers lies in the use of M-matrix theory. In the monotonic range of Peclet numbers, the Petrov-Galerkin method performs well in the sense that oscillatory solutions are not present in the flow. With proper stabilization, finite element matrix equations can be iteratively solved by the Lanczos method, used concurrently with local minimization provided by GMRES(1). The resulting BiCGSTAB iterative solver, supplemented with the Jacobi preconditioner, is implemented in an element-by-element fashion. This gives solutions which are computationally feasible for large-scale flow simulations. The results of two computations are presented in support of the ability of the present finite element model to resolve sharp gradients in the solution. As is apparent from this study is that considerable savings in computer storage and execution time are achieved in adaptive meshes through use of the preconditioned BiCGSTAB iterative solver.