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Building (yet other) bridges between polytopal methods

Simon Lemaire: Thursday, 13th March 2025 at 10:30 Abstract: Within the last 20 years or so, a myriad of novel numerical approaches, capable of accommodating general polytopal meshes, have pop up in the literature. The main purpose of this talk is to tidy up the room, and to build connections, in the context of a model variable diffusion problem, between these different approaches. Our study will focus on skeletal methods. As opposed to plain-vanilla finite volume and discontinuous Galerkin discretizations, skeletal methods essentially attach degrees of freedom to the mesh skeleton. Our study will discriminate between primal and mixed formulations of the problem at hand. Somewhat unsurprisingly, we will see that, at the end of the day, all these approaches fall within only two distinct approximation paradigms.

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Shape optimisation using Lipschitz functions

Philip Herbert: Thursday, 06th March 2025 at 10:30 Abstract: In this talk, we discuss a novel method in PDE constrained shape optimisation.  While it is known that many shape optimisation problems have a solution, finding the solution, or an approximation of the solution, may prove non-trivial.  A typical approach to minimisation is to use a first order method; this raises questions when handling shapes – what is a shape derivative, where does it live?  It happens to be convenient to define the derivatives as linear functionals on $W^{1,\infty}$.  We present an analysis of this in a discrete setting along with the existence of directions of steepest descent.  Several numerical experiments will be considered and extensions discussed.

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Perspectives in structure-preserving numerical schemes

Martin Licht: Thursday, 06th February 2025 at 10:00 Abstract: Structure-preserving numerical methods have had a transformative impact on the numerical analysis of partial differential equations, reproducing the fundamental mathematical structures of numerous partial differential equations exactly at the numerical level. This talk gives an introduction and overview of structure-preserving finite element methods via “finite element exterior calculus” (FEEC) and explores some new directions in the field. FEEC is a comprehensive framework for mixed finite element methods, at the heart of which are finite element differential complexes. This comprehensive theory of mixed finite element methods connects geometry, topology, and classical numerical analysis. We highlight recent research developments, including mixed boundary conditions in FEEC and finite element methods over manifolds, and discuss some future directions in numerical electromagnetism, elasticity, and fluid dynamics.

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Discontinuous Galerkin finite element methods for the control constrained Dirichlet control problem governed by the diffusion equation.

Divay Garg: Thursday, 30th January 2025 at 10:30 Abstract: We utilize a unified discontinuous Galerkin approach to approximate the control constrained Dirichlet boundary optimal control problem using finite element method over simplicial triangulation. The continuous optimality system obtained from this method simplifies the control constraints into a simplified Signorini type problem, which is then coupled with boundary value problems for the state and co-state variables. The symmetric property of the discrete bilinear forms is required in order to derive the discrete optimality system. The main focus is to derive residual based a posteriori error estimates in the energy norm, where we address the reliability and efficiency of the proposed a posteriori error estimator. The suitable construction of auxiliary problems, continuous and discrete Lagrange multipliers, and intermediate operators are crucial in developing a posteriori error analysis. We have also established optimal a priori error estimates in the energy norm for all the optimal variables (state, co-state, and control) under the appropriate regularity assumptions. Theoretical findings are confirmed and illustrated through numerical results on both uniform and adaptive meshes.

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Adaptive homotopy continuation for relative permeability models in reservoir simulation

Peter Moritz von Schultzendorff: Monday, 13th January 2025 at 10:30 Accurate modeling of physical processes requires an appropriate selection of constitutive laws. In physics-based reservoir simulation, constitutive laws, e.g., relative permeabilities are often chosen to be, mathematically speaking, simple functions, not necessarily adhering to physics. The paradigm of hybrid modeling allows the integration of machine learned (ML) constitutive laws. Trained on lab, field, and fine-scale simulation data, ML models represent the underlying physics with high fidelity.Strong nonlinearities in classic (i.e., non-ML) relative permeability have been identified as one of the main sources for convergence issues of nonlinear solvers in reservoir simulation. This issue grows in severance for ML relative permeability models, as their high fidelity to real-world data compromises the mathematically desirable properties of simpler models.In this work, we employ the homotopy continuation (HC) method to recover nonlinear solver robustness for classic relative permeability models. The HC method improves nonlinear solver robustness by first solving a problem with simpler relative permeabilities and then iteratively traversing a solution curve towards the original, more complex problem. To efficiently trace the solution curve, we leverage a posteriori error estimates to design an adaptive HC algorithm that minimizes the total number of solver iterations.We show the current status of our work, both on the theoretical and implementation side, and give an outlook into the application to ML relative permeabilities.

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