Preconditioning spectral element schemes for definite and indefinite problems

Shapira, Yair; Israeli, M.; Sidi, Avram; Zrahia, Uzi

doi:10.1002/(sici)1098-2426(199909)15:5<535::aid-num1>3.0.co;2-r

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…When p increases, however, SE-specific techniques should be used. In particular, we refer to (i) domain decomposition-based preconditioners (Pavarino & Widlund 2000) and (ii) preconditioners based on FD and FE (Shapira et al 1999). AMG methods can be used effectively in conjunction with SE methods, just as for FE.…”

Section: (C ) Optimal Matrix Solution Methodsmentioning

confidence: 99%

Simulation of cardiac electrophysiology on next-generation high-performance computers

Bordas

Carpentieri²,

Fotia³

et al. 2009

Phil. Trans. R. Soc. A.

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Models of cardiac electrophysiology consist of a system of partial differential equations (PDEs) coupled with a system of ordinary differential equations representing cell membrane dynamics. Current software to solve such models does not provide the required computational speed for practical applications. One reason for this is that little use is made of recent developments in adaptive numerical algorithms for solving systems of PDEs. Studies have suggested that a speedup of up to two orders of magnitude is possible by using adaptive methods. The challenge lies in the efficient implementation of adaptive algorithms on massively parallel computers. The finite-element (FE) method is often used in heart simulators as it can encapsulate the complex geometry and small-scale details of the human heart. An alternative is the spectral element (SE) method, a high-order technique that provides the flexibility and accuracy of FE, but with a reduced number of degrees of freedom. The feasibility of implementing a parallel SE algorithm based on fully unstructured all-hexahedra meshes is discussed. A major computational task is solution of the large algebraic system resulting from FE or SE discretization. Choice of linear solver and preconditioner has a substantial effect on efficiency. A fully parallel implementation based on dynamic partitioning that accounts for load balance, communication and data movement costs is required. Each of these methods must be implemented on nextgeneration supercomputers in order to realize the necessary speedup. The problems that this may cause, and some of the techniques that are beginning to be developed to overcome these issues, are described.

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Section: (C ) Optimal Matrix Solution Methodsmentioning

confidence: 99%