Domain decomposition methods: Difference between revisions

Revision as of 08:59, 19 September 2013

Revision as of 07:42, 27 November 2014

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[[Mortar method]]s are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by [[multiple-point constraint]]s.

Domain decomposition methods were popular in the 1990s and early 2000s because they allowed for the parallel solution of partial differential equations on [[parallel computers]] such as [[computer cluster|cluster]], without overly complicated data structures. However, they have since fallen out of favor because they deteriorate with large processor counts. Furthermore, software libraries (such as PETSc and Trilinos) have since become widely available that allow to treat very large scale PDE discretizations at the global level and provide the necessary data transfer to consider just a single, global discretization; they also provide parallel preconditioners that work on the global scale and do not deterioriate too much with the problem size (e.g., [[multigrid method|algebraic multigrid]] methods).

Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.

==Example 1: 1D Linear BVP==