4,532 research outputs found
Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps
Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property
Optimized Schwarz Methods for Maxwell equations
Over the last two decades, classical Schwarz methods have been extended to
systems of hyperbolic partial differential equations, and it was observed that
the classical Schwarz method can be convergent even without overlap in certain
cases. This is in strong contrast to the behavior of classical Schwarz methods
applied to elliptic problems, for which overlap is essential for convergence.
Over the last decade, optimized Schwarz methods have been developed for
elliptic partial differential equations. These methods use more effective
transmission conditions between subdomains, and are also convergent without
overlap for elliptic problems. We show here why the classical Schwarz method
applied to the hyperbolic problem converges without overlap for Maxwell's
equations. The reason is that the method is equivalent to a simple optimized
Schwarz method for an equivalent elliptic problem. Using this link, we show how
to develop more efficient Schwarz methods than the classical ones for the
Maxwell's equations. We illustrate our findings with numerical results
Space Decompositions and Solvers for Discontinuous Galerkin Methods
We present a brief overview of the different domain and space decomposition
techniques that enter in developing and analyzing solvers for discontinuous
Galerkin methods. Emphasis is given to the novel and distinct features that
arise when considering DG discretizations over conforming methods. Connections
and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
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