Numerical Analysis Group Internal Seminar

Tim van Beeck (University of Göttingen) is going to talk about: "An Embedded Trefftz DG Approach to Least-Squares Finite Element Methods"

Least-squares finite element methods (LSFEM) minimise the residual of the PDE directly. This gives discrete problems that are stable by construction, have symmetric positive definite system matrices, and come with a built-in reliable and efficient error estimator. In this talk, we present an embedded Trefftz DG approach to LSFEM. Trefftz methods use basis functions that locally lie in the kernel of the differential operator, which greatly reduces the number of degrees of freedom. The embedded Trefftz DG method obtains these spaces through an element-local embedding into a standard DG space, so no explicit Trefftz basis is needed. We apply the method to a range of linear second-order problems, including Helmholtz-type equations and PDEs in non-divergence form, and show that it keeps the properties above under suitable mesh resolution conditions, while requiring far fewer degrees of freedom than standard DG or first-order least-squares formulations. We conclude with an outlook on nonlinear problems such as the Hamilton-Jacobi-Bellman equation.

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Christian Alber will talk about 'Multiscale Spectral Generalised Finite Elements: Preconditioning and Model Reduction Beyond SPD'

Multiscale PDEs with heterogeneous, highly oscillatory coefficients can be challenging for standard numerical methods. This talk presents a framework for the design and analysis of the Multiscale Spectral Generalised Finite Element Method (MS-GFEM), a partition of unity method using optimal local approximation spaces constructed from eigenproblems. These spaces are closely related to GenEO coarse spaces in domain decomposition.  Under suitable assumptions, we establish nearly exponential local approximation rates and describe their dependence on problem parameters. We also discuss how the resulting coarse spaces can be used in two-level domain decomposition preconditioners. Applications to Helmholtz and Stokes problems illustrate the approach beyond the symmetric positive definite setting, with numerical experiments complementing the analysis.

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