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    Numerical solution of nonlinear finite element problems in elasticity
    (2026)
    Fesefeldt, Eva Lina
    ;
    Le Borne, Sabine
    We consider high-order finite element discretizations of displacement problems on elastic bodies under external loading. In general, the deformation of an elastic body is nonlinear with respect to the applied load. Divergence of the nonlinear solver is prevented by applying the loads in increments. We study a modification of the traditional incremental load step method that exploits the hierarchical finite element discretization. We test the new approach on a benchmark problem in computational mechanics and on a displacement-driven foam pore problem.
    Publicationtype: Doctoral Thesis
      65  61
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    Construction of hierarchical matrices for the preconditioning of the three-dimensional Navier-Stokes equations
    (2025)
    Grams, Jonas David
    ;
    Le Borne, Sabine
    Solving saddle point problems, e.g. obtained from the discretization of the Navier-Stokes equations, with Krylov subspace methods is a difficult task due to often unfavorable (spectral) properties of the system matrix. A popular preconditioning technique for those kinds of systems are block preconditioners relying on approximations to the upper left system matrix block and the so-called Schur complement. These approximations can be obtained by hierarchical matrix LU factorizations but require the explicit computation of an approximate Schur complement. We present improvements in the construction of the required hierarchical matrices leading to a speed-up for the preconditioner set-up.
    Publicationtype: Doctoral Thesis
      98  84
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    A block householder based algorithm for the QR decomposition of hierarchical matrices
    (2025)
    Griem, Vincent Eric
    ;
    Le Borne, Sabine
    This thesis introduces a new approach to compute QR factorizations of hierarchical matrices using block Householder transformations and a newly developed implicit storage scheme for them. The algorithm and the analysis of its numerical cost track low-rank factorizations in intermediate results. Numerical tests on several types of square matrices like 2D Laplacian boundary element matrices and different RBF kernel matrices show a good performance, although the algorithm struggles with rectangular matrices. A version of the H-LU decomposition is also implemented, demonstrating potential benefits.
    Publicationtype: Doctoral Thesis
      51  60
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    Low-rank update of preconditioners for saddle-point systems in fluid flow problems
    (2024)
    Beddig, Rebekka Salome
    ;
    Le Borne, Sabine
    We develop and analyze low-rank updates for preconditioners that are based on a (randomized) low-rank approximation of the error between the identity matrix and the preconditioned matrix. The low-rank updates are applied to different components of a block preconditioner for saddle-point systems that arise in the simulation of the rotating Boussinesq equations. A relaxation parameter improves the initial preconditioner as well as the low-rank updates. We analyze the complexity of the construction and application for different update schemes. An error analysis explains why low-rank corrections may deteriorate a given preconditioner. Numerical experiments illustrate the impact of various parameters on the effectiveness of the low-rank updates.
    Publicationtype: Doctoral Thesis
      98  145
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    Iterative solvers for RBF-FD discretized flow problems
    (2024)
    Leinen, Willi
    ;
    Le Borne, Sabine
    This thesis deals with the numerical solution of partial differential equations via the radial basis function - finite difference (RBF-FD) method and the iterative solution of the arising global linear systems of equations. The focus is on approximation errors and preconditioners based on incomplete LU decompositions, on LU decompositions of hierarchical matrices, and on auxiliary stiffness matrices with smaller stencil sizes.
    Publicationtype: Doctoral Thesis
      112  344
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    Efficient numerical treatment of aggregation integrals in multivariate population balance equations
    (2020)
    Ahrens, Robin
    ;
    Le Borne, Sabine
    This work develops efficient numerical methods for aggregation integrals in multivariate population balance equations on a uniform tensor grid. These are based on the fast Fourier transform and the tensor-train format for the efficient storage of the resulting data structure. The inverse problem of kernelestimation from discrete in time data is additionally addressed. All results are underlined with numerical simulations.
    Publicationtype: Doctoral Thesis
      388  644
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    Efficient and accurate evaluation of aggregation integrals in population balance equations
    (2017)
    Shahmuradyan, Lusine
    ;
    Le Borne, Sabine
    Applications of population balance equations can be found in many chemical production processes, e.g, crystallization and precipitation of pharmaceutical materials. In population balance equations the aggregation process is described by source and sink aggregation integrals. The source aggregation integral is usually numerically expensive to evaluate having quadratic complexity. In this thesis we focus on the efficient and accurate evaluation of aggregation integrals, reducing this complexity to almost linear. We also include extensive numerical tests for different setups, leading to (heuristic) strategies for the choice of the discretization parameters.
    Publicationtype: Doctoral Thesis
      475  450