Vincent Darrigrand

This research aims to produce error indicators for the pre-asymptotical regime of highly unstable problems such as the Helmholtz equation and the convection-dominated diffusion problem in the context of p-GOA.

The classical approach provides an upper bound of the error in the Quantity of Interest (QoI) by the element-wise product of the direct and adjoint problems errors. This representation is used as error indicators to drive adaptive processes. However, for non-SPD problems, these indicators may not lead to convergent adaptive process due, for instance, to the pollution error in the Helmholtz case. We propose to compute the Riesz representation of the adjoint problem's error. Thus, we bound the error in the QoI by the element-wise product of the direct problem error and the Riesz representation. We studied the effect on a p-adaptive process applied to Helmholtz and convection-dominated diffusion problems. We observed an error stabilization phenomenon, and we were able to improve the adaptive process behavior.

We refer to this publication, this one, and this one. I built my Ph.D. Thesis with those results.

We implement our method on top of Zander et al. hierarchical data-structure. Our main contribution resides in the adaptive strategy. For a given mesh, we aim at returning the coarsest discretization possible without overly increasing the error.

In this paper, we describe the algorithm and apply it to Symmetric Positive Definite problems (manufactured solutions for a Laplace equation). The adaptive algorithm is the following:

1. Start with an initial coarse mesh
2. Refine it (user-defined refinement, eg. global h-refinements and/or global p-refinements)
3. Perform the coarsening step
4. Return to step 1 until the mesh is fine enough and the error below the threshold

The unrefinement step proceeds as follows:

1. Compute the solution
2. Identify the removable basis functions
3. Evaluate their contribution
4. Remove the least contributing ones

We illustrate the process in the following animation:

To build the hp-coarsening step, we need a criteria to know if we should proceed to h- or p-unrefinements. For a given element,

• If its contribution is small compare to the average of contributions then proceed to p-unrefinement,

• if the contribution of its siblings are also small, then perform an h-unrefinement instead.

We are interested in saddle-point problems (linear elastic with MPC) from structural mechanics and provided by EDF. Historically, code_aster employs the double lagrangian approach to ensure that the saddle-point problem is SPD to use Cholesky factorization. However, this approach notably increased the system’s size, making it even more difficult to resolve via direct solvers. Hence, we need to find an adapted iterative solver for this situation. The idea proposed here explores the use of the generalized Golub-Kahan Bidiagonalization (GKB) iterative method on the augmented Lagrangian system to solve those problems.

The significant contribution is proof that for a sufficiently large augmentation parameter, the number of iterations of the GKB algorithm should be independent of the problem size. However, since we rely on eigenvalues' computation to determine the parameter values, we have to proceed experimentally.

To validate this result, we constructed a systematic study on test cases by increasing the difficulty both in terms of problem size and conditioning. We showed that, for with the parameter set to the 1-norm of the (1,1)-block, we obtain satisfactory convergence. The most challenging problem is the discretization of a containment building. We showed that even for this realistic industrial problem, we could achieve the convergence of the GKB within a few iterations. Finally, we showed that GKB scalability is limited by the (1,1)-block solver scalability.

Those results are published in this article

In the framework of a post-doc within the European Project EoCoE II, we are interested in linear elastic problems encountered in applications such as wind farms where structural mechanics are coupled with fluid dynamics. The need for scalable solvers is critical. I am investigating the use of Domain Decomposition Methods (DDM) coupled with direct solvers. We particularly investigate the use of some of their novel features such as the Block-Low Rank (BLR) feature and advanced tree multithreading to improve the complexity of the overall solver.