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Questions tagged [domain-decomposition]

A parallel algorithm design approach in which the data is divided into pieces and then computations are associated with the data. This contrasts to 'functional decomposition', in which tasks or computations are divided first, then data is associated to them.

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15
votes
5answers
3k views

What is the advantage of multigrid over domain decomposition preconditioners, and vice versa?

This is mostly aimed for elliptic PDEs over convex domains, so that I can get a good overview of the two methods.
12
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3answers
310 views

In what application cases are additive preconditioning schemes superior to multiplicative ones?

In both domain decomposition (DD) and multigrid (MG) methods, one may compose the application of the block updates or coarse corrections as either additive or multiplicative. For pointwise solvers, ...
11
votes
3answers
1k views

Best Methodologies for Managing a Mesh in Parallel Finite Element Computation?

I am currently developing a domain decomposition method for the solution of the scattering problem. Basically I am solving a system of Helmholtz BVPs iteratively. I discretize the equations using ...
6
votes
1answer
142 views

Compability conditions in domain decomposition methods

Suppose we want to solve the Poisson equation $\Delta u = f$ on a domain $\Omega$ with Dirichlet boundary conditions. One possible way to do is by a domain decomposition method. There is a condition ...
6
votes
1answer
209 views

Domain decomposition w/Lagrange multipliers

I'm looking at FEM discretizations of $$u_i - \Delta u_i = f$$ for $u_1, u_2$ on subdomains $\Omega_1, \Omega_2$ with interface $\Gamma$. A Neumann-Neumann transmission condition can be formulated by ...
5
votes
3answers
820 views

implicit vs. explicit domain decomposition methods

I've been working on a finite element code on unstructured methods, which I've parallelized using the Schur complement method. Here's a summary of how I did it: Assign each triangle of the mesh to a ...
5
votes
1answer
136 views

Effect of subdomain topologies on overlapping additive Schwarz?

Is there a reference on the effect of subdomain topology on performance of the overlapping additive Schwarz method for (high order) finite elements? For example, taking subdomains to be vertex ...
5
votes
0answers
118 views

Load balancing/partitioning with unknown weights

For a grid-based numerical simulation, I am looking for a load balancing/partitioning algorithm that not only distributes my grid elements, but also determines (approximates) their respective weights. ...
4
votes
0answers
238 views

Is it possible to predict the null space of a structure from contributing elements null spaces?

I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-...
3
votes
1answer
128 views

What is the global problem in the two-level additive Schwarz?

The two-level additive Schwarz method (additive Schwarz with a coarse space correction) is often written like this: $$ \mathbf{v} = \sum_{i=0}^N \mathbf{R}_i^T \mathbf{A}^{-1}_i\mathbf{R}_i\mathbf{w} $...
3
votes
1answer
132 views

nonoverlapping domain decomposition

I solved a simple test example by overlapping domain decomposition. The problem domain is a rectangular that is decomposed to two domains. So the value on the intersection boundary is guessed at the ...
3
votes
2answers
230 views

Steklov-Poincaré operator for overlapping domain decomposition

For non-overlapping domain decomposition methods for elliptic problems there is an associated Steklov-Poincaré definite positive operator defined on the interface, allowing a direct computation of the ...
3
votes
0answers
144 views

Alternating Schwarz does not converge without Dirichlet conditions on physical boundaries

Note: Thanks to comments, I realized that I have two problems, each which can be described more clearly on its own. This revised question covers the first. I would like to solve $$ -\Delta u - 1=0,\...
2
votes
2answers
171 views

How to precondition FEM problems using domain decomposition?

Let's say that I have a FEM code which yields the following problem: $$ \mathbf{A}\mathbf{x} = \mathbf{b}. $$ In order to solve this more efficiently with an iterative method, I would like to ...
2
votes
2answers
959 views

Mesh domain decompositions / mesh partitioning

I have some experience with mpmetis from METIS. It is pretty good software which offers unstructured mesh grid partitioning. But obtained results always minimize edgecuts or total communication volume....
2
votes
2answers
347 views

Direct Solver and domain decomposition

In finite element method, in order to use MPI, we need to decompose the domain into sub-domains first. Then my question is whether we can solve each sub-domain using a direct solver? Of course, unlike ...
2
votes
1answer
84 views

why overlapping technique can accelerate the additive/multiplictive Schwarz

Overlapping technique can make each subdomain contain more nodes, and the overlapped subdomains are nonlonger disjoint, is it taking the average value of the multiple nodes as the result. After ...
2
votes
0answers
110 views

scalable parallel mesh/amr on unstructured grid

I am trying to code a scalable parallel AMR for unstructured grid. There seems to be three approaches for this a) Store some global grid info on each processor and partition with parmetis (The ...
1
vote
1answer
185 views

Efficient evaluation of $BQ^{-1}B^T$ (Domain Decomposition Implementation)

I have implemented a schur complement domain decomposition method for solving large scale problems in Matlab. It works well but I think there should be some alternative approaches to implement some ...
1
vote
1answer
78 views

Partitioning SPD matrix with METIS to preserve block SPD-ness

I am using the METIS to partition a matrix and then using domain decomposition to solve the subdomains in parallel using the Restricted Additive Schwarz method. I am currently trying to solve some ...
1
vote
0answers
67 views

understanding Domain Decomposition with example

I am new in Domain Decomposition method. I am started to solve $-\Delta u = f$ in $\Omega$ and $u = 0$ on $\partial\Omega$. From that I get in $\Omega _1$ $$\begin{bmatrix}4&-1\\-1&4\end{...
1
vote
0answers
32 views

Metis: how to use and tutorial recommendation

I am new to METIS and trying to use it in my fortran code. I read the manual online. But still, I am not sure about how to implement it my code. I tried the test cases in the graphs directory. For ...
1
vote
0answers
62 views

overlapping additive schwarz [closed]

I am solving 1D laplace problem discretized with finite differences (3-point stencil). I would like to use additive Schwarz method in classical form: $U_{k+1}=U_{k}+M^{−1} r_k,$ where $r_k=F−A U_k$...
1
vote
0answers
69 views

Non-overlaping Domain decomposition - assemble of Laplacian

I am dealing with following 2-dimensional problem in the unit square domain $S_2$ $$- \Delta u (x,y) = f \ \text{in} \ S_2, \hspace{1.5cm} u(x,y) = 0 \ \text{on} \ \partial S_2$$ where $f$ is ...
1
vote
0answers
165 views

Enforcing continuity conditions in pseudospectral domain decomposition methods for time dependent PDEs

I have a partial differential equation of the form $$ \frac{d}{dt}f(x,t) = \Theta(x) f(x,t) \qquad \Theta(x) \sim \left[\frac{d^2}{dx^2} + k^2(x)\right] $$ subject to $f(x,t=0) = f_0(x)$, and $f(x=0,t)...
0
votes
1answer
139 views

how to partition a graph(matrix) into subdomains with different sizes

i am studying the solver for PageRank problems which drived from the web link graph. I have tried using METIS to divided the matrix into subdomains, but METIS can only produce subdomains with nearly ...
0
votes
1answer
987 views

Domain Decomposition with PETSc

Does anyone have any experience on Domain Decomposition using PETSc library? I have used PETSc for creating my vectors and matrix within my C++ code. I also used KSP to solve the linear system. I ...