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December 4, 2007
Abstract
The degenerate isotropic boundary value problem –∇(ω 2 ( x )∇ u ( x, y )) = f ( x, y ) on the unit square (0, 1) 2 is considered in this paper. The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned conjugate gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α ≠ 1. The preconditioning operation requires O( N ) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.
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December 4, 2007
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In this second part of our two-part article, we present and discuss the corresponding numerical results from implementations of the numerical algorithms described in the first part. With these results, we observed that • operator splitting applied to the associated time-dependent problem is suitable for solving only the first eigenproblem, • among those tried, the perturbation and arclength continuation approach was the sole effective and robust approach for solving higher eigenproblems, • on the eigenproblems for which (undamped or damped) Newton's method converged, it was without question the most efficient.
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December 4, 2007
Abstract
This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn's inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. variations of the Re number on unstructured meshes. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.