Program
The presentations take place in the seminar room 2.417 (RUD 25).
Schedule for Week 1
Schedule for Week 2
| Day | Time | Speaker | Title |
|---|---|---|---|
| Mon 07/11 | 01:15 pm | Dietmar Gallistl (Jena) | Computational lower bounds of the Maxwell eigenvalues |
| Mon 07/11 | 03:15 pm | Xuefeng Liu (Niigata) | Rigorous eigenfunction computation and its application to computer-assisted proof |
| Tue 08/11 | 09:15 am | Sophie Puttkammer (Berlin) | Direct guaranteed lower eigenvalue bounds with quasi-optimal adaptive mesh-refinement |
| Tue 08/11 | 11:15 am | Ngoc Tien Tran (Jena) | A finite element method for uniformly elliptic linear PDEs of second order in nondivergence form |
| Thu 10/11 | 09:15 am | Roland Maier (Jena) | Semi-explicit time discretization schemes for elliptic-parabolic problems |
| Thu 10/11 | 11:15 am | Benedikt Gräßle (Berlin) | A hybrid high-order method for guaranteed lower eigenvalue bounds |
Schedule for Week 3
| Day | Time | Speaker | Title |
|---|---|---|---|
| Thu 17/11 | 09:15 am | Zhiqiang Cai (Purdue) | Adaptive neural network method |
| Thu 17/11 | 03:15 pm | Zhiqiang Cai (Purdue) |
Least-squares neural network (LSNN) method for hyperbolic conservation laws
(Different room: lecture hall 1.013 in Rudower Chaussee 25) |
Abstracts
| Speaker: | Joscha Gedicke (Bonn) |
|---|---|
| Title: | A posteriori error analysis for the TDNNS finite element method |
| This is ongoing work. We will discuss the features of the TDNNS finite element method and the current progress of the derivation of a residual type a posteriori error estimator, it's challenges and open questions. | |
| Speaker: | Rekha Khot (IITB) |
| Title: | Quasi-optimal nonconforming VEM for the biharmonic equation |
|
The lowest-order nonconforming virtual element
extends the Morley triangular element to polygons
for the approximation of the weak solution u ∈ V
:= H²₀(Ω) to the biharmonic equation. Two
nonconforming virtual element spaces have been
introduced in [1, 4] for H³ regular solutions,
while a medius analysis in [3] allows minimal
regularity. In this talk, we will discuss an
abstract framework (cf. [2]) with two hypotheses
(H1)-(H2) for a unified stability and a priori
error analysis of at least two different discrete
spaces (even a mixture of those). A smoother J
allows rough source terms F ∈ V* = H⁻²(Ω). The a
priori and a posteriori error analysis circumvents
any trace of second derivatives by a computable
conforming companion operator J : Vₕ → V from the
nonconforming virtual element space Vₕ. The
operator J is a right-inverse of the interpolation
operator and leads to optimal error estimates in
piecewise Sobolev norms without any additional
regularity assumptions on u ∈ V. As a smoother the
companion operator modifies the discrete
right-hand side and then allows a quasi-best
approximation. An explicit residual-based a
posteriori error estimator is reliable and
efficient up to data oscillations. Numerical
examples display the predicted empirical
convergence rates for uniform and optimal
convergence rates for adaptive mesh-refinement.
References
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| Speaker: | Zhaonan Dong (INRIA) |
| Title: | Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the biharmonic problem |
| We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C¹-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H² space and a generalized Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behavior of the estimator is investigated through numerical examples in two and three dimensions. | |
| Speaker: | Benedikt Gräßle (Berlin) |
| Title: | Stabilization-free reliable and efficient a posteriori error control for HHO |
| The established a posteriori error analysis of the hybrid high-order methods (HHO) treats the stabilization as part of the error and as part of the error estimator. But it follows from [Ern--Zanotti, 2020] that the stabilization is in fact efficient. This leads to reliable and efficient explicit residual-based a posteriori error estimates for the error in the piecewise energy norm (up to data oscillations). This talk presents recent work proving that the original and the VEM inspired stabilizations for HHO are locally equivalent and derive two classes of stabilization-free guaranteed upper bounds (GUB). Numerical evidence in a Poisson model problem supports that the GUB lead to realistic upper bounds and the associated adaptive mesh-refining algorithm recovers the optimal convergence rates in computational benchmarks. | |
| Speaker: | Andreas Veeser (Milano) |
| Title: | Accurate error bounds for nonconforming Galerkin methods |
| An error bounds is typically an inequality between two seminorms and may call such an error bound accurate whenever the two seminorms are equivalent. We illustrate this viewpoint with various examples, focusing on the context of nonconforming Galerkin methods. | |
| Speaker: | Joscha Gedicke (Bonn) |
| Title: | A posteriori error analysis for symmetric mixed Arnold-Winther FEM |
| The development of mixed finite element methods for linear elasticity with strongly imposed symmetry has been a long standing problem until the beginning of this century. Surprisingly for a mixed method, nodal stress degrees of freedom are necessary in order to fulfill the strong symmetry. This interesting mixed finite element also poses some difficulties for the derivation of residual based a posteriori error estimators. In a first attempt we make use of the residual a posteriori error estimator techniques for weakly symmetric stresses introducing an auxiliary approximation of the skew-symmetric gradient via a postprocessing. The second version then makes fully use of the imposed symmetry of the stress approximations utilising integration by parts twice and a suitable decomposition into tangential-tangential and normal-normal parts, similarly to the residual a posteriori error analysis for plate problems. The presented postprocessing technique can also be used to compute directly an a posteriori error bound. This is demonstrated for the Stokes problem and the Stokes eigenvalue problem, where a postprocessing of the eigenvalues is proven to lead to improved convergence rates of the eigenvalue errors. | |
| Speaker: | Pietro Zanotti (Pavia) |
| Title: | A quasi-optimal nonconforming discretization for the stationary Biot equations in poroelasticity |
| In the theory of poroelasticity, the Biot equations model the flow of a fluid inside a linear elastic porous medium. We are concerned with the space discretization of the equations after a time semi-discretization with the backward Euler scheme. A typical difficulty encountered in this task is the robustness with respect to various material parameters. We deal with it by observing that the problem is uniformly stable, irrespective of all parameters, in a suitable nonsymmetric variational setting. Guided by this result, we design a novel nonconforming discretization, which employs Crouzeix-Raviart and discontinuous elements. We prove that the proposed discretization is quasi-optimal and robust in a parameter-dependent norm. | |
| Speaker: | Miriam Schönauer, Andreas Schröder (Salzburg) |
| Title: | An optimal AFEM for elastoplasticity with the application of the axioms of adaptivity |
| In this talk a proof of optimal convergence of an AFEM for elastoplasticity with combined linear kinematic and isotropic hardening is discussed. The proof applies the axioms of adaptivity. The verification of the axioms in this setting draws from an already existing result on optimal convergence for AFEM for elastoplasticity. These two approaches to proving optimal convergence are then compared to highlight differences and similarities between their methodologies. | |
| Speaker: | Philipp Bringmann (Berlin) |
| Title: | Convergence analysis and numerical comparison of adaptive least-squares finite element methods |
| Due to the built-in a posteriori error control, the least-squares finite element methods (LSFEMs) are a favourable choice for adaptive mesh-refining algorithms. Convergence results have been established for various adaptive LSFEMs in the literature. First, the built-in error estimator leads to Q-linear convergence in an adaptive algorithm with collective marking. Second, an alternative residual-based error estimator and a separate marking strategy with data approximation even guarantee optimal convergence rates for the error in the natural underlying norm. Third, collective marking with the alternative error estimator provides optimal convergence rates in a weaker norm. An experimental comparison of all three adaptive algorithms confirms these findings. The first part of this talk outlines the state-of-the-art for the convergence analysis of adaptive LSFEMs. The second part investigates the choice of the parameters in the marking and refinement strategies as well as the performance of the adaptive algorithms. | |
| Speaker: | Gerhard Starke (Duisburg-Essen) |
| Title: | Stress-Based Adaptivity for Quasi-Variational Inequalities Associated with Frictional Contact |
| The stress-based formulation of elastic contact with Coulomb friction in the form of a quasi-variational inequality is investigated. Weakly symmetric stress approximations are constructed using a finite element combination on the basis of Raviart-Thomas spaces of next-to-lowest order. An error estimator is derived based on a displacement reconstruction and proved to be reliable under certain assumptions on the solution formulated in terms of a norm equivalence in the trace space. Numerical results illustrate the effectiveness of the adaptive refinement strategy for a Hertzian frictional contact problem in the compressible as well as in the incompressible case. | |
| Speaker: | Dietmar Gallistl (Jena) |
| Title: | Computational lower bounds of the Maxwell eigenvalues |
| A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [2013] for the Laplace operator. The main tool is the computation of an explicit upper bound to the error of the Galerkin projection. The error is split in two parts: one part is controlled by a hypercircle principle and an auxiliary eigenvalue problem. The second part requires a perturbation argument for the right-hand side replaced by a suitable piecewise polynomial. The latter error is controlled through the use of the commuting quasi-interpolation by Falk--Winther and computational bounds on its stability constant. This situation is different from the Laplace operator where such a perturbation is easily controlled through local Poincaré inequalities. | |
| Speaker: | Xuefeng Liu (Niigata) |
| Title: | Rigorous eigenfunction computation and its application to computer-assisted proof |
|
We are concerned with the rigorous computation of
eigenfunction (or eigen-space) of differential
operators. Upon the setting of eigenvalue
problems, several methods are proposed to give
rigorous and efficient error estimation for the
approximate eigenfunctions. The rigorous
eigenfunction computation is applied to solving
shape optimization problems. By explicitly
evaluating the Hadamard shape derivative with
guaranteed computation of both eigenvalues and
eigenfunctions, one can adopt the
computer-assisted proof to solve the shape
optimization problems. For example, for the
constant C(K) related to the Crouzeix–Raviart
interpolation ΠCR over triangle K:
it is proved that the optimal value of C(K) ≈ 0.1893 is taken when K is a regular triangle. |
|
| Speaker: | Roland Maier (Jena) |
| Title: | Semi-explicit time discretization schemes for elliptic-parabolic problems |
| This talk is about semi-explicit time-stepping schemes for coupled elliptic-parabolic problems as they arise, for instance, in the context of poroelasticity. If the coupling between the equations is rather weak, such schemes may be applied. Their main advantage is that they decouple the equations and therefore allow for faster computations. Theoretical convergence results are presented that rely on a close connection of the semi-explicit schemes to partial differential equations that include delay terms. Numerical experiments confirm these results. | |
| Speaker: | Sophie Puttkammer (Berlin) |
| Title: | Direct guaranteed lower eigenvalue bounds with quasi-optimal adaptive mesh-refinement |
| Guaranteed lower eigenvalue bounds (GLB) are of high relevance. A post-processing for nonconforming FEM computes GLB, but the maximal mesh-size enters as a global parameter and can cause significant underestimation for adaptive mesh-refinement. This talk presents a modified hybrid high-order (HHO) method (m = 1) as well as an extra-stabilized nonconforming Crouzeix-Raviart (m = 1) and Morley (m = 2) FEM. These methods compute direct GLB for the m-Laplace operator in that a specific smallness assumption on the maximal mesh-size guarantees that the computed k-th discrete eigenvalue is a lower bound for the k-th Dirichlet eigenvalue. This talk shows striking numerical evidence for the superiority of a new adaptive eigensolver that motivates the convergence analysis. For the extra-stabilized nonconforming methods (a generalization of) known abstract arguments entitled as the axioms of adaptivity verify the convergence of the GLB towards a simple eigenvalue with optimal rates. | |
| Speaker: | Ngoc Tien Tran (Jena) |
| Title: | A finite element method for uniformly elliptic linear PDEs of second order in nondivergence form |
| This talk presents a finite element scheme for the approximation of strong solutions to uniformly elliptic linear PDEs of second order in nondivergence form. The main tool will be the Alexandrov-Bakelman-Pucci maximum principle that provides control over the maximum norm as well as the local W2,n Sobolev norm of the error. | |
| Speaker: | Zhiqiang Cai (Purdue) |
| Title: | Adaptive neural network method |
|
(joint work with Jingshuang Chen and Min Liu)
The first part of this talk is an introduction to neural network as a new class of approximating functions and its application to numerical partial differential equations. The second part of this talk is on our recent work [1, 2, 3] on adaptively designing a nearly optimal neural network architecture for a given approximation problem with a prescribed accuracy. References
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| Speaker: | Zhiqiang Cai (Purdue) |
| Title: | Least-squares neural network (LSNN) method for hyperbolic conservation laws |
|
(joint work with Jingshuang Chen and Min Liu)
Solutions of nonlinear hyperbolic conservation laws (HCLs) are often discontinuous due to shock formation; moreover, locations of shocks are a priori unknown. This presents a great challenge for traditional numerical methods because most of them are based on continuous or discontinuous piecewise polynomials on fixed meshes. As an alternative, by employing a new class of approximating functions, neural network (NN), recently we proposed the least-squares neural network (LSNN) method for solving HCLs. The LSNN method shows a great potential to sharply capture shock without oscillation or smearing; moreover, its degrees of freedom are much less than those of mesh-based methods. Nevertheless, current iterative solvers for the LSNN discretization are computationally intensive and complicated. In this talk, I will present our recent work [1, 2, 3] on the LSNN for solving linear and nonlinear scalar HCLs. References
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