## (Français) 𝐻² conforming virtual element discretization of nondivergence form elliptic equations

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Simulation for the Environment: Reliable and Efficient Numerical Algorithms

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Guillaume Bonnet: Thursday, 21st November at 11:00 The numerical discretisation of elliptic equations in nondivergence form is notoriously challenging, due to the lack of a notion of weak solutions based on variational principles. In many cases, there still is a well-posed variational formulation for such equations, which has the particularity of being posed in 𝐻², and therefore leads to a strong solution. Galerkin discretizations based on this formulation have been studied in the literature. Since 𝐻² conforming finite elements tend to be considered impractical, most of these discretizations are of discontinuous Galerkin type. On the other hand, it has been observed in the virtual element literature that the virtual element method provides a practical way to build 𝐻² conforming discretizations of variational problems. In this talk, I will describe a virtual element discretization of equations in nondivergence form. I will start with a simple linear model problem, and show how the 𝐻² conformity of the method allows for a particularly simple well-posedness and error analysis. I will then discuss the extension to equations with lower-order terms and with Hamilton-Jacobi-Bellman type nonlinearities, and present some numerical results.

Jørgen S. Dokken: Thursday, 10th October at 11:00 The FEniCS project was started in 2003 as a collaboration between ToyotaTechnological Institute at Chicago, The University of Chicago and Chalmers University of Technology. The vision of the project was to create an open-source project that included the automation of modelling, optimization and discretization of differential equations. The plan was to create a generalized and efficient framework that was easy to use. With this goal the project expanded from being a pure C++ code (DOLFIN) to being a combined C++/Python framework for solving PDEs. One of the most notable outcomes of the project is the Unified Form Language, a domain specific language for representing PDEs in a variational form using computational symbolic algebra. In this talk, I will go through some of the historical context of the FEniCS project and its evolution into DOLFINx (DOI: 10.5281/zenodo.10447666). A presentation of the core components will highlight the newest developments and the extensibility of the framework.

André Harnist: Tuesday, 15th October at 11:00 We design a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the usual energy difference that depend locally on the nonlinearity and improve the established bound. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.

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Weifeng Qiu: Wednesday, 2nd October at 11:00 Abstract: In this presentation. I will discuss DG type methods for incompressible MHD and Maxwell’s transmission eigenvalues, and ALE for two-phase incompressible Naiver-Stokes flow. I will also discuss possible further works along these directions.

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Gregor Gantner: Thursday, 17th October at 11:00 Abstract: In this talk, we discuss stable space-time FEM-BEM couplings [1] to numerically solve parabolic transmission problems on the full space and a finite time interval.The couplings are based on the space-time FOSLS [2,3] in the interior and the space-time BEM [4] in the exterior.In particular, we demonstrate coercivity of the couplings under certain restrictions and validate our theoretical findings by numerical experiments. REFERENCES [1] T. Führer, G. Gantner, and M. Karkulik, Space-time FEM-BEM couplings for parabolic transmission problems, Preprint, arXiv:2409.14449 (2024). [2] T. Führer and M. Karkulik, Space–time least-squares finite elements for parabolic equations, Comput. Math. Appl., 92 (2021), pp. 27–36. [3] G. Gantner and R. Stevenson, Further results on a space-time FOSLS formulation of parabolic PDEs, ESAIM Math. Model. Numer. Anal., 55 (2021), pp. 283–299. [4] M. Costabel, Boundary integral operators for the heat equation, Integral Equations Operator Theory, 13 (1990), pp. 498–552.

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Carsten Carstensen: Monday, 10th September at 11:00 Abstract: The popular (piecewise) quadratic schemes for the fourth-order plate bending problems based on triangles are the nonconforming Morley finite element, two discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. The first part of the presentation discusses recent applications to the linear bi-Laplacian and to semi-linear fourth-order problems like the stream function vorticity formulation of incompressible 2D Navier-Stokes problem and the von Kármán plate bending problem. The role of a smoother is emphasised and reliable and efficient a posteriori error estimators give rise to adaptive mesh-refining strategies that recover optimal rates in numerical experiments. The last part addresses recent developments on adaptive multilevel Argyris finite element methods. The presentation is based on joint work with B. Gräßle, (Humboldt) and N. Nataraj (IITB, Mumbai) partly reflected in the references below. REFERENCES [1] C. Carstensen, B. Gräßle,, and N. Nataraj. Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation, J. Numer. Math., volume 32, pp. 77–109, 2024, arXiv:2310.05648. [2] C. Carstensen, B. Gräßle,, and N. Nataraj. A posteriori error control for fourth-order semilinear problems with quadratic nonlinearity, SIAM J. Numer. Anal., volume 62, pp. 919–945, 2024. [3] C. Carstensen, Jun Hu. Hierarchical Argyris finite element method for adaptive and multigrid algorithms, Comput. Methods Appl. Math., volume 21, pp. 529–556, 2021. [4] C. Carstensen, N. Nataraj. A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides, Comput. Methods Appl. Math., volume 21, pp. 289–315, 2021. [5] C. Carstensen, N. Nataraj, G.C. Remesan, D. Shylaja. Lowest-order FEM for fourth-order semi-linear problems with trilinear nonlinearity, Numerische Mathematik 154, pp. 323–368, 2023. [6] C. Carstensen, N. Nataraj. Lowest-order equivalent nonstandard finite element methods for biharmonic plates, ESAIM: Mathematical Modelling and Numerical Analysis, 56(1),…