Related papers: From delayed minimization to the harmonic map heat…
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an…
In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates,…
We study a general convergence theory for the analysis of numerical solutions to the magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (= 2; 3). First, we…
We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables…
We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the…
We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear…
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to…
Large optimal transport problems can be approached via domain decomposition, i.e. by iteratively solving small partial problems independently and in parallel. Convergence to the global minimizers under suitable assumptions has been shown in…
The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we…
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite…
This thesis addresses the question of stability of systems defined by differential equations which contain nonlinearity and delay. In particular, we analyze the stability of a well-known delayed nonlinear implementation of a certain…
Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…
We consider singularly perturbed gradient flows in Hilbert spaces, driven by a time-dependent, nonconvex, and nonsmooth energy, and address the convergence of their solutions to curves of critical points of the driving energy functional.…
We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of…
We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class…
We present a finite element discretisation to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensuring continuity of…
Optimizing problems in a distributed manner is critical for systems involving multiple agents with private data. Despite substantial interest, a unified method for analyzing the convergence rates of distributed optimization algorithms is…
In this paper we study the harmonic map heat flow problem for a radially symmetric case. The corresponding partial dfferential equation plays a key role in many analyses of harmonic map heat flow problems. We consider a basic discretization…
This article presents a detailed introduction to density-based topology optimisation of fluid flow problems. The goal is to allow new students and researchers to quickly get started in the research area and to skip many of the initial…
Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy…