In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between the adjacent subdomains. A corse problem with one or fiew unknows per subdomain is used to further coordinate the solution between the subdomains globally.
We consider the following laplacian boundary value problem
where
The schwartz overlapping multiplicative algorithm with dirichlet interface conditions for this problem at iteration is given by
where and
.
Let , the error at
iteration relative to the exact solution, the convergence rate is given by
find such that
We consider the following laplacian boundary value problem
where and
is the dirichlet boundary value.
The schwartz overlapping multiplicative algorithm with dirichlet interface conditions for this problem on two subdomains and
at
iteration is given by
The numerical results presented in the following table correspond to the partition of the global domain in two subdomains
and
and the following configuration:
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Geometry |
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Nomber of iterations | ![]() | ![]() |
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11 | 2.52e-8 | 2.16e-8 |
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Isovalues of Solution in 2D |
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We consider at the continuous level the Dirichlet-to-Neumann(DtN) map on , denoted by DtN
. Let
where is a bounded domain of
(d=2 or 3), and
it border,
is a positive diffusion function which can be discontinuous, and
. The eigenmodes of the Dirichlet-to-Neumann operator are solutions of the following eigenvalues problem
To obtain the discrete form of the DtN map, we consider the variational form of . let's define the bilinear form
,
With a finite element basis , the coefficient matrix of a Neumann boundary value problem in
is
A variational formulation of the flux reads
So the variational formulation of the eigenvalue problem reads
Let be the weighted mass matrix
The compact form of is
Assembly of the right hand side
Assembly of the left hand side
solve the eigenvalue problem
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Three eigenmodes |
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These numerical solutions correspond to the following configuration :