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Hybrid high-order methods for the wave equation in first-order form

Rekha Khot: Thursday, 20th March 2025 at 10:30 Abstract: In this talk, we will discuss the approximation of the acoustic wave equation in its first-order Friedrichs formulation using hybrid high-order (HHO) methods, proposed and numerically investigated in [Burman-Duran-Ern, 2022]. We first look at energy-error estimates in the time-continuous setting and give several examples of interpolation operators: the classical one in the HHO literature based on L2-orthogonal projections and others from, or inspired from, the hybridizable discontinuous Galerkin (HDG) literature giving improved convergence rates on simplices. The time-discrete setting is based on explicit Runge-Kutta (ERK) schemes in time combined with HHO methods in space. In the fully discrete analysis, the key observation is that it becomes crucial to bound the consistency error in space by means of the stabilization seminorm only. We formulate three abstract properties (A1)-(A3) to lead the analysis. Our main result proves that, under suitable CFL conditions for second- and third-order ERK schemes, the energy error converges optimally in time and quasi-optimally in space, with optimal rates recovered on simplicial meshes. The abstract foundations of our analysis should facilitate its application to other nonconforming hybrid methods such as HDG and weak Galerkin (WG) methods.

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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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