7,484 research outputs found
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
A Nonlinear Multigrid Steady-State Solver for Microflow
We develop a nonlinear multigrid method to solve the steady state of
microflow, which is modeled by the high order moment system derived recently
for the steady-state Boltzmann equation with ES-BGK collision term. The solver
adopts a symmetric Gauss-Seidel iterative scheme nested by a local Newton
iteration on grid cell level as its smoother. Numerical examples show that the
solver is insensitive to the parameters in the implementation thus is quite
robust. It is demonstrated that expected efficiency improvement is achieved by
the proposed method in comparison with the direct time-stepping scheme
A parallel multigrid solver for multi-patch Isogeometric Analysis
Isogeometric Analysis (IgA) is a framework for setting up spline-based
discretizations of partial differential equations, which has been introduced
around a decade ago and has gained much attention since then. If large spline
degrees are considered, one obtains the approximation power of a high-order
method, but the number of degrees of freedom behaves like for a low-order
method. One important ingredient to use a discretization with large spline
degree, is a robust and preferably parallelizable solver. While numerical
evidence shows that multigrid solvers with standard smoothers (like Gauss
Seidel) does not perform well if the spline degree is increased, the multigrid
solvers proposed by the authors and their co-workers proved to behave optimal
both in the grid size and the spline degree. In the present paper, the authors
want to show that those solvers are parallelizable and that they scale well in
a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF)
for the financial support through the DK W1214-04, while the second author
was supported by the FWF grant NFN S117-0
Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations
Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method
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