We consider the solution of second order elliptic PDEs in $\R^d$ with inhomogeneous Dirichlet data by means of an $h$-adaptive FEM with fixed polynomial order $p\in\N$. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an $H^{1/2}$-stable projection, for instance, the $L^2$-projection for $p=1$ or the Scott-Zhang projection for general $p\ge1$. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each $H^{1/2}$-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the $L^2$- and Scott-Zhang projection conclude the work.
One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called $ϕ$-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries…
On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators
Inria（法国国家数字与工程科学研究所）；Univ. Lille（里尔大学）；Laboratoire Paul Painlevé（保罗·庞加莱实验室）；Inria Paris（法国国家数字与工程科学研究所巴黎分部）；Institute for Numerical Simulation, University of Bonn（波恩大学数值模拟研究所）；University of Twente（特文特大学）；Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）
We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use $H^{-1}$ loads. We prove that any bounded $H^{-1}$ projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Clément quasi-interpolator. We prove that this Clément …
An extra-stabilised Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplace Dirichlet eigenvalues. The smallness assumption $\min\{λ_h,λ\}h_{\max}^{4}$ $\le 184.9570$ in $2$D (resp. $\le 21.2912$ in $3$D) on the maximal mesh-size $h_{\max}$ makes the computed $k$-th discrete eigenvalue $λ_h\le λ$ a lower eigenvalue bound for the $k$-th Dirichlet eigenvalue $λ$. This holds for multiple and clusters of eigenvalues and serves for the localisation of the bi-Laplacian Dirichlet eigenvalues in particular for c…
A posteriori upper and lower bounds are derived for the linear finite element method (FEM) for the Helmholtz equation with large wave number. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size $h$ in the preasymptotic regime, which is first observed by Babuška, et al. for an one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the corresponding elliptic projection of the exact solution, an adaptive algorithm is proposed and its c…
WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains
School of Science, Harbin Institute of Technology, Shenzhen, P. R. China（哈尔滨工业大学深圳校区）；School of Science, Harbin Institute of Technology, Shenzhen, Guangdong（哈尔滨工业大学深圳校区）；Institute of Structural Mechanics, Bauhaus-Universität Weimar（魏玛 Bauhaus 大学结构力学研究所）
Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs
In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(Ω)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(Ω)^{d,d}$-norms, the existence of minimizers and optimality systems of the regula…
This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter optimization methods. In particular, we present several different adaptive RB approaches that can be used in an error-aware trust-region framework for progressive construction of a surrogate model used during a certified outer optimization loop. In addition, we elaborate on several different enhancements for the trust-region reduced basis (TR-RB) algorithm and…
