Home Archive by category "News"

Category: News

A view into the development of the FEniCS project over two decades

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.

Continue reading

Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems.

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.

Continue reading

Space-time FEM-BEM couplings for parabolic transmission problems

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.

Continue reading

Adaptive computation of fourth-order problems

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),…

Continue reading

Computable reliable bounds for Poincaré–Friedrichs constants via Čech–de-Rham complexes

Martin Licht: Thursday, 25th April at 11:00 Abstract: We derive computable and reliable upper bounds for Poincaré–Friedrichs constants of classical Sobolev spaces and, more generally, Sobolev de-Rham complexes. The upper bounds are in terms of local Poincaré–Friedrichs constants over subdomains and the smallest singular value of a finite-dimensional operator that is easily assembled from the geometric setting. Thus we reduce the computational effort when computing the Poincaré–Friedrichs constant of finite de-Rham complexes, and we provide computable reliable bounds even for the original Sobolev de-Rham complex. The reduction to a finite-dimensional system uses diagram chasing within a Čech–de-Rham complex.

Continue reading