Prof. Carsten Carstensen
Contact: Tim Stiebert (stiebert@math.hu-berlin.de)
The seminar talks usually take place on Tuesday at 11 c.t..
Humboldt-Universität zu Berlin, Institut für Mathematik
Rudower Chaussee 25, 12489 Berlin
House 2, Floor 4, Room 2.417
| Date | Time | Talk by | Title | Room |
|---|---|---|---|---|
| 13.10.2026 | 11:15 | Ma Rui (Beijing Institute of Technology) |
Adaptive H(divdiv) mixed finite element methods for the Kirchhoff-Love plate bending problem | 2.417 |
| 20.10.2026 | 10:15 | Dirk Praetorius (TU Wien) |
Rate-optimal adaptive FEM with dynamic Dörfler marking | 2.417 |
| 23.10.2026 | 13:15 | Max Naumann (HU Berlin) |
Estimator comparison for mixed boundary conditions in 3D Poisson problem | online |
| 27.10.2026 | 11:15 | Tim Stiebert (HU Berlin) |
Guaranteed lower inf-sup bounds and applications to general second-order elliptic PDE | 2.417 |
| 03.11.2026 | 11:15 | Lara Theallier (HU Berlin) |
TBA | 2.417 |
| 18.11.2026 | 15:15 | Qiu Weifeng (City University of Hong Kong) |
An Interface Green's Function Framework for Complete Discrete W1,∞ Analysis of Discontinuous Galerkin Methods | 2.417 |
| Speaker: | Ma Rui (Beijing Institute of Technology) |
|---|---|
| Title: | Adaptive H(divdiv) mixed finite element methods for the Kirchhoff-Love plate bending problem |
| This talk presents a reliable and efficient residual-based a posteriori error analysis for the symmetric H(divdiv) mixed finite element method for the Kirchhoff-Love plate bending problem with mixed boundary conditions. The key ingredient lies in the construction of boundary-condition preserving complexes at both continuous and discrete levels. Numerical examples confirm the effectiveness of the a posteriori error estimator and demonstrate the optimal convergence rate under adaptive refinements. | |
| Speaker: | Dirk Praetorius (TU Wien) |
| Title: | Rate-optimal adaptive FEM with dynamic Dörfler marking |
|
Meanwhile convergence and optimal convergence rates of adaptive FEM (AFEM) are fairly well understood,at least, for linear elliptic PDEs. The current state of the art on the standard AFEM loop
SOLVE -- ESTIMATE -- MARK -- REFINE can be paraphrased as follows: First, any Dörfler marking parameter 0 < θ <= 1 guarantees R-linear convergence of the estimator sequence. Second, if θ is sufficiently small, then the estimator sequence decays at rate s > 0 with respect to the number of degrees of freedom, where s is as large as possible in terms of certain nonlinear approximation classes. In practice, one observes that also „moderate“ θ lead to optimal convergence rates. In our talk, we propose a slight modification of the adaptive loop, where the Dörfler marking is chosen dynamically (and decays logarithmically to 0). If a convergence rate s > 0 is possible, then the extended algorithm guarantees convergence at any rate 0 < t < s. Eventually, this proves that the extended AFEM loop is essentially optimal. | |
| Speaker: | Qiu Weifeng (City University of Hong Kong) |
| Title: | An Interface Green's Function Framework for Complete Discrete W1,∞ Analysis of Discontinuous Galerkin Methods |
|
Pointwise error analysis of discontinuous Galerkin (DG) methods for the Poisson equation has received considerable attention over the past two decades. However, on convex polyhedral domains, existing analyses can only establish nearly (with a logarithmic factor) optimal error estimates in the L∞ norm and optimal error estimates in the broken W1,∞ seminorm for several DG methods. Since numerical solutions obtained by DG methods may have jumps across mesh interfaces, and the broken W1,∞ seminorm does not control these jumps, a complete discrete W1,∞ theory for DG methods for the Poisson equation on convex polyhedral domains has remained unavailable. Here, by a complete discrete W1,∞ theory, we mean that the maximum-norm of the broken gradient and the maximum norm of the interface jumps are estimated simultaneously.
In this paper, we develop an interface Green's function framework for the complete discrete W1,∞ analysis of DG methods on convex polyhedral domains. The proposed framework introduces new interface Green's functions that represent the jumps of numerical solutions of DG methods across mesh interfaces. Its central analytical ingredient is a new local energy estimate for these Green's functions, obtained by exploiting a cancellation between neighboring discrete delta functions. This local energy estimate differs fundamentally from existing Green's function estimates and enables us to derive maximum-norm estimates for the jumps of numerical solutions across mesh interfaces without introducing additional logarithmic factors. Via the proposed framework, we establish optimal error estimates in the complete discrete W1,∞ norm for the symmetric interior penalty DG method, the hybrid mixed DG method, and two representative hybridizable DG methods for the Poisson equation in convex polyhedral domains. In particular, we derive optimal maximum-norm estimates for interface jumps, numerical fluxes, and broken gradients. We further extend the framework to the C0 interior penalty method for the biharmonic equation on convex polygonal domains and obtain maximum-norm estimates for the jumps of the normal derivative. The interface Green's function framework is sufficiently general to be applicable to a broad class of nonconforming methods and is valid on general polyhedral meshes. | |