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Development and analysis of a block-preconditioner for the phase-field crystal equation. (English) Zbl 1320.82059

Summary: We develop a preconditioner for the linear system arising from a finite element discretization of the phase-field crystal (PFC) equation. The PFC model serves as an atomic description of crystalline materials on diffusive time scales and thus offers the opportunity to study long time behavior of materials with atomic details. This requires adaptive time stepping and efficient time-discretization schemes, for which we use an embedded Rosenbrock scheme. To resolve spatial scales of practical relevance, parallel algorithms are also required, which scale to large numbers of processors. The developed preconditioner provides such a tool. It is based on an approximate factorization of the system matrix and can be implemented efficiently. The preconditioner is analyzed in detail and shown to speed up the computation drastically.

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65Z05 Applications to the sciences
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
82D25 Statistical mechanics of crystals
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