Abstract
Spectral element method is in fact an extension of spectral method inspired by the concepts used in finite element method. In this paper, for solving linear second order partial differential equations with Dirichlet boundary conditions on irregular regions, a new type of mix spectral element method will be examined. We use polar coordinates transformation for our proposed method. As a new technique, to avoid ill-conditioning of the coefficient matrix of the obtained system, we use the condition of the equality of the right and left derivatives (with respect to $$ \theta $$ and $$ r $$ ) with the help of the information related to the adjacent element, except when these points are boundary points themselves, for which the boundary condition is specified. To solve the obtained system of the linear equations, a numerical method on the base of preconditioned approach is applied. Ultimately, the presented method will be compared with the finite element method. The numerical results indicate the efficiency of our method.
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