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Open Access
January 1, 2010
Abstract
This paper gives an overview of adaptive discretization methods for linear second-order hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkin-type methods for spatial as well as temporal discretization, which also include variants of the Crank-Nicolson and the Newmark finite difference schemes. The adaptive choice of space and time meshes follows the principle of \goaloriented" adaptivity which is based on a posteriori error estimation employing the solutions of auxiliary dual problems.
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Open Access
January 1, 2010
Abstract
We consider the conforming of finite element approximations of reactiondiffusion problems. We propose new a posteriori error estimators based on H(div)- conforming finite elements and equilibrated fluxes. It is shown that these estimators give rise to an upper bound where the constant is one in front of the indicator, up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh and the local variation of the coefficients. We further analyze the convergence of an adaptive algorithm. The reliability and efficiency of the proposed estimators are confirmed by various numerical tests.
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Open Access
January 1, 2010
Abstract
The paper deals with the numerical computation of a crack problem posed on microstructural heterogeneous materials containing multiple phases in the microstructure. The failure of such materials is a natural multi-scale effect since cracks typically nucleate in regions of defects on the microscopic scale. The modeling strategy for solving the crack problem concerns simultaneously the macroscopic and microscopic models. Our approach is based on an efficient combination of the homogenization technique and the mesh superposition method (s-version of the finite element method). The homogenized model relies on a double-scale asymptotic expansion of the displacement field. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh arbitrarily on the global continuous mesh. The crack is treated by the local mesh and the homogenized material model is considered on the global mesh. Numerical experiments for problems on biomorphic microcellular ceramic templates with porous microstructures of different materials constituents are presented.
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Open Access
January 1, 2010
Abstract
The present paper discusses the parallelization of the robust multigrid technique (RMT) and the possible way of applying this to unstructured grids. As opposed to the classical multigrid methods, the RMT is a trivial method of parallelization on coarse grids independent of the smoothing iterations. Estimates of the minimum speed-up and parallelism efficiency are given. An almost perfect load balance is demonstrated in a 3D illustrative test. To overcome the geometric nature of the technique, the RMT is used as a preconditioner in solving PDEs on unstructured grids. The procedure of auxiliary structured grids generation is considered in details.
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Open Access
January 1, 2010
Abstract
In this paper, under certain conditions, we show that the solution of the semidiscrete form of a semilinear heat equation with a variable reaction is quenched in a finite time and estimate its semidiscrete quenching time. We also show that the semidiscrete quenching time converges to the continuous one when the mesh size goes to zero. In the same way, an analogous study has been investigated taking into account the discrete form of the above problem. Finally, we present some computational results.
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Open Access
January 1, 2010
Abstract
We present a second-order finite difference method for obtaining a solution of a second order two-point boundary value problem subject to Sturm's boundary conditions. We use equidistant discretization points, and the discretization of the differential equation at an interior point is based on just two evaluations of the function. Numerical examples are considered and the convergence of the proposed method is proved computationally.