3,126 research outputs found
Cross-Points in Domain Decomposition Methods with a Finite Element Discretization
Non-overlapping domain decomposition methods necessarily have to exchange
Dirichlet and Neumann traces at interfaces in order to be able to converge to
the underlying mono-domain solution. Well known such non-overlapping methods
are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and
optimized Schwarz methods. For all these methods, cross-points in the domain
decomposition configuration where more than two subdomains meet do not pose any
problem at the continuous level, but care must be taken when the methods are
discretized. We show in this paper two possible approaches for the consistent
discretization of Neumann conditions at cross-points in a Finite Element
setting
Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions
Optimized Schwarz Waveform Relaxation methods have been developed over the
last decade for the parallel solution of evolution problems. They are based on
a decomposition in space and an iteration, where only subproblems in space-time
need to be solved. Each subproblem can be simulated using an adapted numerical
method, for example with local time stepping, or one can even use a different
model in different subdomains, which makes these methods very suitable also
from a modeling point of view. For rapid convergence however, it is important
to use effective transmission conditions between the space-time subdomains, and
for best performance, these transmission conditions need to take the physics of
the underlying evolution problem into account. The optimization of these
transmission conditions leads to a mathematically hard best approximation
problem of homographic type. We study in this paper in detail this problem for
the case of linear advection reaction diffusion equations in two spatial
dimensions. We prove comprehensively best approximation results for
transmission conditions of Robin and Ventcel type. We give for each case closed
form asymptotic values for the parameters, which guarantee asymptotically best
performance of the iterative methods. We finally show extensive numerical
experiments, and we measure performance corresponding to our analysisComment: 42 page
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
Closed form optimized transmission conditions for complex diffusion with many subdomains
Optimized transmission conditions in domain decomposition methods have been
the focus of intensive research efforts over the past decade. Traditionally,
transmission conditions are optimized for two subdomain model configurations,
and then used in practice for many subdomains. We optimize here transmission
conditions for the first time directly for many subdomains for a class of
complex diffusion problems. Our asymptotic analysis leads to closed form
optimized transmission conditions for many subdomains, and shows that the
asymptotic best choice in the mesh size only differs from the two subdomain
best choice in the constants, for which we derive the dependence on the number
of subdomains explicitly, including the limiting case of an infinite number of
subdomains, leading to new insight into scalability. Our results include both
Robin and Ventcell transmission conditions, and we also optimize for the first
time a two-sided Ventcell condition. We illustrate our results with numerical
experiments, both for situations covered by our analysis and situations that go
beyond
Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping for heterogeneous problems
We design and analyze a Schwarz waveform relaxation algorithm for domain
decomposition of advection-diffusion-reaction problems with strong
heterogeneities. The interfaces are curved, and we use optimized Robin or
Ventcell transmission conditions. We analyze the semi-discretization in time
with Discontinuous Galerkin as well. We also show two-dimensional numerical
results using generalized mortar finite elements in space
A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks
We present a meshless Schwarz-type non-overlapping domain decomposition
method based on artificial neural networks for solving forward and inverse
problems involving partial differential equations (PDEs). To ensure the
consistency of solutions across neighboring subdomains, we adopt a generalized
Robin-type interface condition, assigning unique Robin parameters to each
subdomain. These subdomain-specific Robin parameters are learned to minimize
the mismatch on the Robin interface condition, facilitating efficient
information exchange during training. Our method is applicable to both the
Laplace's and Helmholtz equations. It represents local solutions by an
independent neural network model which is trained to minimize the loss on the
governing PDE while strictly enforcing boundary and interface conditions
through an augmented Lagrangian formalism. A key strength of our method lies in
its ability to learn a Robin parameter for each subdomain, thereby enhancing
information exchange with its neighboring subdomains. We observe that the
learned Robin parameters adapt to the local behavior of the solution, domain
partitioning and subdomain location relative to the overall domain. Extensive
experiments on forward and inverse problems, including one-way and two-way
decompositions with crosspoints, demonstrate the versatility and performance of
our proposed approach
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