A two-level finite element method and its application to the Helmholtz equation

Prof. Leopoldo Franca

Department of Mathematics, University of Colorado at Denver Germany

First, we revisit the Galerkin finite element method using piecewise polynomials enriched with special functions that we denote by {\it residual-free bubble functions}. Partitioning our domain into a mesh of elements, the residual-free bubble functions are defined to be as rich as possible within an element. In other words, these functions are assumed to satisfy strongly the partial differential equations in the interior of the element, up to the contribution of the piecewise polynomial functions. In addition, they are also assumed to satisfy a homogeneous Dirichlet condition on the element boundary. The residual-free-bubbles represent the unresolvable part of the solution, whereas the piecewise polynomials are the resolvable part for the given mesh. This decomposition of the solution into a piecewise polynomial plus residual-free bubbles produces the exact solution of linear differential equations in the one-dimensional case. Furthermore, by inspecting the method after we eliminate the residual-free bubbles, various successful discretization schemes are unveiled, such as upwinding for advective-diffusive equations, mass lumping for a model of the parabolic heat transfer equation, selective reduced integration with adjustment of coefficients for the deflection of a Timoshenko beam, etc. In higher dimensions the computation of the residual-free bubbles becomes a major task, in that only in limited situations (such as rectangular elements) one can employ classical analytical tools to get the exact solution within each element. In this talk we introduce a two-level finite element method consisting of a mesh for discretization and a submesh. The Galerkin method with piecewise polynomials augmented with residual-free bubble functions is used in the mesh and the submesh is employed for approximating the computations of residual-free bubble basis functions. The submesh is defined in the interior of each element, where a ``tricky'' numerical method is used to approximate the residual-free bubble functions. Once these are determined, the effect of the residual-free bubbles on the piecewise polynomial part of the solution can be calculated to find the solution of the Galerkin method in the original mesh. This method does not suffer from drawbacks of having to solve analytically partial differential equations in the element interior, and therefore it is suitable for any irregular mesh, used in practice in finite element computations. Furthermore, it provides a systematic framework to generate discretizations. We will present preliminary computations of this method for approximating the Helmholtz equation in two dimensions.