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PRECONDITIONING DISCRETIZATIONS OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
, 2009
"... This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be c ..."
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This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be considered. In particular, parameter dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several examples and models which have been discussed in the literature previously. However, here each example is discussed with reference to a more unified abstract approach.
BLOCK PRECONDITIONERS FOR FULLY IMPLICIT RUNGEKUTTA SCHEMES APPLIED TO THE BIDOMAIN EQUATIONS
"... Key words: Block preconditioners RungeKutta methods, orderoptimal methods Abstract. Recently, the authors presented different block preconditioners for implicit RungeKutta discretization of the heat equation. The preconditioners were block Jacobi and block GaussSeidel preconditoners where the bl ..."
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Key words: Block preconditioners RungeKutta methods, orderoptimal methods Abstract. Recently, the authors presented different block preconditioners for implicit RungeKutta discretization of the heat equation. The preconditioners were block Jacobi and block GaussSeidel preconditoners where the blocks reused existing preconditioners for the implicit Euler discretization of the same equation. In this paper we will introduce similar block preconditioners for the implicit RungeKutta discretization of the Bidomain equation. We will, by numerical experiments, show the properties of the preconditoners, and that higherorder RungeKutta discretization of the Bidomain equation may be superior to lowerorder in some cases. 1
BLOCK PRECONDITIONERS FOR COUPLED PHYSICS PROBLEMS∗
"... Abstract. Finite element discretizations of multiphysics problems frequently give rise to blockstructured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of wellknown block factorizations [21, 16] and apply these to the diffusiv ..."
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Abstract. Finite element discretizations of multiphysics problems frequently give rise to blockstructured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of wellknown block factorizations [21, 16] and apply these to the diffusive portion of the bidomain equations and the Bénard convection problem. An abstract generalized eigenvalue problem allows us to give applicationspecific bounds for the real parts of eigenvalues for these two problems. This analysis is accompanied by numerical calculations with several interesting features. One of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes. For Bénard convection, we observe meshindependent convergence with reasonable robustness with respect to physical parameters, and offer some preliminary parallel scaling results on a multicore processor via MPI. 1. Introduction. Numerical
Computing and Information STABILITY OF TWO TIMEINTEGRATORS FOR THE ALIEVPANFILOV SYSTEM
"... Abstract. We propose a secondorder accurate method for computing the solutions to the AlievPanfilov model of cardiac excitation. This twovariable reactiondiffusion system is due to its simplicity a popular choice for modeling important problems in electrocardiology; e.g. cardiac arrhythmias. The ..."
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Abstract. We propose a secondorder accurate method for computing the solutions to the AlievPanfilov model of cardiac excitation. This twovariable reactiondiffusion system is due to its simplicity a popular choice for modeling important problems in electrocardiology; e.g. cardiac arrhythmias. The solutions might be very complicated in structure, and hence highly resolved numerical simulations are called for to capture the fine details. Usually the forward Euler timeintegrator is applied in these computations; it is very simple to implement and can be effective for coarse grids. For finescale simulations, however, the forward Euler method suffers from a severe timestep restriction, rendering it less efficient for simulations where high resolution and accuracy are important. We analyze the stability of the proposed secondorder method and the forward Euler scheme when applied to the AlievPanfilov model. Compared to the Euler method the suggested scheme has a much weaker timestep restriction, and promises to be more efficient for computations on finer meshes. Key Words. reactiondiffusion system, implict RungeKutta, electrocardiology 1.
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"... This survey paper is based on three talks given by the second author at the London Mathematical Society ..."
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This survey paper is based on three talks given by the second author at the London Mathematical Society