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## FETI with one Lagrange Multiplier

The previous ideas are now formalized for the general case where we have Ns > 2 subdomains. Using any non-overlapping partition of the domain into subdomains (s), s = 1; : : : ; Ns. We define the interface [ = (@ (s) \ @ (q)) 1 s;q Ns In the FETI method the discrete flux, noted , is the unknown vector defined in the nodes along the interface. The jump in the solutions of the local Neumann problems is the gradient of the condensed interface problem.

With the previous notation, the discretization of the local Neumann problems in some subdomain (s), can be written as K(s)x(s) = f (s) + t(s)T B(s)T (1.28).

where K(s) are the local stiﬀness matrices; f (s) the right-hand side vectors; t(s) 2 M#(@ (s)) #( (s)) are trace operators which extracts boundary degrees of freedom from any subdomain (s). Finally, B(s) 2 Mdim( ) dim(@ (s)) are discrete assembling matrices which connects pairwise degrees of freedom on the inter-face.

### Local preconditioner

The coarse grid smoothing performed by the zero energy fields projector P gives a convergence rate independent upon the number of subdomains, but this is not enough to have a convergence rate that is also independent upon the mesh size. Is necessary the use of a preconditioner, which for the FETI method is one of the “Dirichlet” type.

Consider t(s) the trace or restriction operator on the local interface of subdo-main (s). Then the contribution of this subdomain to the condensed interface operator, defined in (1.32), is B(s)t(s)K(s)+ t(s)T B(s)T.

#### New FETI as a hybrid between one and two Lagrange methods

Now we are ready to present a new method that arises “directly” if we think of the properties of each previous method and the fact that both share similarities in their implementations. The main idea is to mix the good convergence and speed of the FETI-1LM method with the robustness of the FETI-2LM and apply it in more complex scenarios that usually we have to face in real life applications.

To show this, we can see in Figure 1.8 a contact problem coming from the scheme model of a rolling bearing. These kinds of problems can be easily extended to other contact schemes where we want to compute the interactions between objects of diﬀerent materials, that can be as diﬀerent as steel and rubber. The problems with heterogeneous contact boundary conditions imposed with the penalty method [81] are the type of situations that can make the discretization matrices of the model (and consequently the FETI operator) very ill-conditioned.

In this configuration a natural interface will be one that divides the two materials, if this is the case, then the FETI-1LM method with the extended preconditioner may be appropriate to handle the heterogeneities (even here the convergence is not assured if the diﬀerences are “extreme”). However, this is not always the case when you do an automatic partitioning of the global mesh, or if, for example, we use diﬀerent discretizations that lead to diﬀerent types of meshing, then we will have the case of a non-conforming mesh. Other cases were anisotropic materials needs to be modeled, will also give bad numerical features on the interface. For those problems, the FETI-1LM will not assure the usual fast convergence. The FETI-2LM method is, in general, a more robust method, due to the formulation of two independent Lagrange multipliers, that can overcome these numerical issues. If we think now in the interfaces that are completely contained in one of the two material, or in general do not present numerical issues, the regular FETI method will be more suited as we will get a faster convergence.

Using one or the other will gives us either fast or accurate result, but we can try to get both of these good features if we choose properly which method to apply in every part of the interface. If we manage to make the choice that better suits every local interface, then we will most likely have a hybrid method with both good qualities.

**Dirichlet preconditioner for two subdomains**

We note that the following parts, up to the definition of the S-FETI method, are heavily based on the works of Daniel J. Rixen [60], [61], [62], and can be considered as classic results. However, we think that is necessary to recall them, to fully understand how the S-FETI method works. We will start by revisiting the Dirichlet preconditioner at its basic, for a two subdomain case. The notation, in this part, will be analogous to the previous chapter.

The objective of a preconditioner for the PCPG Algorithm 5 in FETI is to build in every iteration p a correction to the Lagrange multipliers. This rectification is based on an interface compatibility error, i.e., a jump denoted as g (we drop the notation gp for simplicity.). In the case of an elasticity problem this is a jump in the displacement, but depending on the problem this may change.

**Table of contents :**

Abstract

Acknowledgements

Contents

List of Tables

List of Figures

Introduction

Contributions of this thesis

Iterative Methods

Krylov Methods

Conjugate Gradient

ORTHODIR

Parallelization of Krylov methods

**1 Hybrid FETI method **

1.1 Basic FETI method

1.1.1 Model problem and discretization

1.1.2 FETI with one Lagrange Multiplier

1.1.3 Local preconditioner

1.1.4 FETI Algorithms

1.2 FETI with two Lagrange multipliers

1.2.1 FETI-2LM

1.2.2 Arbitrary mesh partition

1.2.3 Optimal Interface Boundary Conditions

1.3 New FETI as a hybrid between one and two Lagrange methods

1.3.1 Development

1.3.2 Extension to a general problem

1.3.3 Preconditioner

1.3.4 Implementation

1.4 Numerical results

1.4.1 Two material bar

1.4.2 Contact Problem

1.5 Conclusion

**2 Block FETI methods **

2.1 Introduction and preliminarities

2.1.1 Dirichlet preconditioner for two subdomains

2.1.2 Consistent Preconditioners

2.1.3 Simultaneous FETI

2.1.4 The algorithm

2.1.5 Cost and implementation of S-FETI

2.2 Sorting search directions in S-FETI

2.2.1 Linear dependence in block FETI directions

2.2.2 Cholesky factorization with complete pivoting

2.2.3 Diagonalization of the search directions block

2.3 Memory usage in S-FETI

2.3.1 New sparse storage

2.3.2 Reconstruction of search directions

2.3.3 Implementation details and exploitable parallelism

2.4 Numerical results

2.4.1 S-FETI basics

2.4.2 Decomposition of WTFW

2.4.3 S-FETI with sparse storage

2.4.4 General comparison

2.5 Conclusion

**3 Block strategies as a preconditioner **

3.1 Introduction and preliminaries

3.1.1 Method of Conjugate Directions

3.1.2 Flexible Conjugate Gradient

3.2 FETI with recursive preconditioner

3.2.1 One direction from S-FETI block

3.2.2 Linear combination of directions from block

3.3 Numerical results

3.3.1 Storage of single direction

3.3.2 Storage of reduced directions

3.4 Conclusion

**4 FETI-2LM with enlarged search space **

4.1 Introduction

4.1.1 The FETI-2LM method

4.2 The Block-2LM Algorithm

4.3 Implementation and cost of the method

4.4 Numerical results

4.4.1 Block-2LM vs 2LM

4.5 Conclusion

Conclusion and perspectives

**Bibliography **