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December 4, 2007
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This paper is on the efficient solution of linear systems arising in discretizations of second order elliptic PDEs by a generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We propose an efficient preconditioner by using auxiliary (fictitious) space techniques and an additive preconditioner for the auxiliary space problems. We also prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh parameters.
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December 4, 2007
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In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operator splitting method applied to the corresponding time-dependent problem. For solving higher eigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by ‘jumping’ to a point on the unperturbed solution branch from a ‘nearby’ point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinear eigenproblem.
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December 4, 2007
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We complete the analysis of our a posteriori error estimators for the time-dependent Stokes problem in R d , d = 2 or 3. Our analysis covers non-conforming finite element approximation (Crouzeix–Raviart's elements) in space and backward Euler's scheme in time. For this discretization, we derived in part I of this paper [J. Numer. Math. (2007) 15, No. 2, 137–162] a residual indicator, which uses a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. In this second part we prove some analytical tools, and derive the lower and upper bounds of the spatial estimator.
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December 4, 2007
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In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –( au ′)′ = f with piecewise-constant coefficient a ( x ), discretized by the hp -FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a ( x ) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.