# FETI based Domain Decomposition Methods

- Parallel and numerical scalability
- FETI, TotalFETI, Hybrid-FETI, FETI-DP
- Usage

Finite element tearing and interconnecting (FETI) domain decomposition methods transform an original large problem to the sequence of smaller problems which can be solved separately and thus exploit the facilities of modern supercomputers. The main idea of FETI methods is to decompose the original subdomain into non-overlapping subdomains. The continuity of the solution between the subdomains is then enforced by Lagrange multipliers. After the elimination of the primal variables the original problem is reduced to a smaller relatively well conditioned dual problem. The figure below shows the part of a forklift gearbox decomposed into 64 subdomains by open-source software METIS. Originally, the FETI methods were introduced to the solution of linear problems. Nowadays, we are able to solve efficiently difficult non-linear problems such as contact problems including friction, problems with material and geometrical nonlinearities and shape optimization with contact.

## Parallel and numerical scalability

Important property of the FETI methods is their scalability. FETI based solvers are able to efficiently utilize the architecture of parallel computers. We say that an algorithm is numerically scalable if the cost of the solution is nearly proportional to the number of unknowns, and it enjoys parallel scalability if the time required for the solution can be reduced nearly proportionally to the number of available processors or processor cores.

## FETI, TotalFETI, Hybrid-FETI, FETI-DP

During the last 20 years several variants of FETI methods were introduced. Let us mention some of them and describe the differences. The FETI method (FETI-1) was originally introduced (by Farhat-Roux) for the numerical solution of the engineering problems described by elliptic partial differential equations. Original domain is decomposed to non-overlapping subdomains. The continuity of the solution between the subdomains is then enforced by Lagrange multipliers. After the elimination of the primal variables the original problem is reduced to a smaller relatively well conditioned dual problem. Local stiffness matrices of the so called “floating” subdomains are singular and so it is necessary to work with pseudoinverses. To avoid the problems with the “floating” subdomains the FETI-DP method were later introduced. In this case the subdomains are connected in several nodes and so the local problems are positive definite. Dual problem resulting from the elimination of primal variables can be solved by PCG (preconditioned conjugate gradients) algorithm.

In our variant Total FETI (TFETI) the subdomains are totally separated as in FETI-1. The Lagrange multipliers enforce the continuity between the subdomains and, moreover, the Dirichlet boundary condition as well. Thanks to that, all local problems are singular (in the case of elasticity) with kernels known a-priori. As a result, local stiffness matrices can be efficiently regularized by standard Cholesky decomposition algorithm for regular matrices.

## Usage

- structural mechanics, thermodynamics, fluid problems
- contact problems, time dependent problems
- nonlinear problems – plasticity, hyperelasticity
- shape optimization