Symmetrization of matrix technique in a finite element context
This short paper presents an interesting symmetrization of matrix technique which can be useful in a finite element context: the authors start by giving two illustrative examples associated with: - the dispersion/diffusion equation; - the Navier-Stokes equations governing the steady-state flow of an incompressible Newtonian fluid. In both cases, a discretization using a standard Galerkin weighted residual procedure and, occasionally, an iterative solution scheme, leads to the solution of a linear system of the following form: \([K]\{a\}=\{f\}\) in which [K] is an unsymmetric \(n\times n\) matrix, \(\{\) \(a\}\) the unknown variables and \(\{\) \(f\}^ a \)column vector. In the current finite element context, the matrix [K] is in fact the summation of element matrices: \([K]=\sum^{e=m}_{e=1}[k]_ e\). Then, the authors propose a symmetrization method at the element level, that means upon each \([k]_ e\). The resulting matrix equation is then solved using either a front- based solver for symmetric matrices or any other appropriate technique. This method results in a significant reduction in core requirement. It can be used to solve both linear and nonlinear systems of equations. But it is very useful mainly for nonlinear problems, where the coefficient matrix already contains updated values of the variables. And, some limitations appear when an unknown approaches zero. Two numerical examples based on convection/diffusion problem on the one hand, and on a laminar flow problem on the other hand, illustrate the efficiency of the method. Perhaps, in forthcoming works, it should be interesting that the authors delimitate more precisely the convergence properties and the limitations of this method.
significant reduction in core requirement
scientific article; zbMATH DE number 3970055