On the solution of algebraic systems for serendipity FEM (Q2730690)

The objective in this paper is to find out the main features of four (two-level) iterative methods, proposed in order to solve linear systems that arise when using serendipity finite elements methods (FEM) for numerically solving a model Dirichlet boundary value problem. The elements used are rectangular with bilinear basis functions corresponding to the vertex nodes and cubic basis functions at the mid-edge points. The matrix structure that arises is conveniently exploited, and fast iterative solvers are constructed which combine efficient preconditioners with the defect correction approach. NEWLINENEWLINENEWLINEThe resulting computational cost is comparable to the one generated by a plain bilinear FEM. The defect correction can be accelerated by means of the conjugate gradient method. For this case, the paper gives results that allow to estimate the number of necessary iterations in order to reduce the norm of the residue to less than a small factor times the initial residue, the estimate depending on this factor. As a consequence, also the complexity can be estimated a priori. NEWLINENEWLINENEWLINEThe paper includes numerical experiments with a particular example that illustrate the accuracy of the different approaches. For instance, the serendipity FEM with uniform grid produces an \(O(h^4)\) approximation, whereas the bilinear FEM yields only second order approximation. More generally, the high accuracy of the serendipity system for a uniform grid (\(O(h^3)\) or even \(O(h^4)\)) enables one to employ a coarse mesh. NEWLINENEWLINENEWLINEAlso, a modified deferred correction approach yields good convergence when the anisotropy coefficient is varied. This approach is shown to be superior to the other three proposed in the paper in the isotropic case, because it requires many fewer operations per iteration.