Related papers: Adaptive hp-Refinement for Spectral Elements in Nu…
This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic…
We consider the standard adaptive finite element loop SOLVE, ESTIMATE, MARK, REFINE, with ESTIMATE being implemented using the $p$-robust equilibrated flux estimator, and MARK being D\"orfler marking. As a refinement strategy we employ…
Physical systems commonly exhibit spatially varying complexity, presenting a significant challenge for neural PDE solvers. While Graph Neural Networks can handle the irregular meshes required for complex geometries and boundary conditions,…
In this work, we propose an adaptive spectral element algorithm for solving nonlinear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer-Gauss points combined with very accurate and stable…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
Material properties such as permeability fields in heterogeneous porous media are often represented as discontinuous, piecewise constant data tied to a given spatial discretization. Such representations are inherently mesh-dependent,…
This research presents the development of an innovative algorithm tailored for the adaptive sampling of residual points within the framework of Physics-Informed Neural Networks (PINNs). By addressing the limitations inherent in existing…
A flexible spectral code for the study of general relativistic magnetohydrodynamics is presented. Aiming at investigating the physics of slowly rotating magnetized compact stars, this new code makes use of various physically motivated…
This article deals with the adaptive and approximative computation of the Lam\'e equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM).…
Adaptive mesh refinement (AMR) offers a practical solution to reduce the computational cost and memory requirement of numerical simulations that use computational meshes. In this work, we introduce a novel smart methodology for adaptive…
Many problems in science and engineering can be rigorously recast into minimizing a suitable energy functional. We have been developing efficient and flexible solution strategies to tackle various minimization problems by employing finite…
A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…
We propose an adaptive scheme of broadening the discrete spectral data from numerical renormalization group (NRG) calculations to improve the resolution of dynamical properties at finite energies. While the conventional scheme overbroadens…
A new gradient-based adaptive sampling method is proposed for design of experiments applications which balances space filling, local refinement, and error minimization objectives while reducing reliance on delicate tuning parameters. High…
Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property…
In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of…
We compare two different codes for simulations of cosmological structure formation to investigate the sensitivity of hydrodynamical instabilities to numerics, in particular, the hydro solver and the application of adaptive mesh refinement…
Image segmentation is an inherently ill-posed problem and thus requires regularization in order to limit the search space to reasonable solutions. A majority of segmentation methods integrates these regularization terms in one way or the…
This article describes a new, fully adaptive Particle-Multiple-Mesh numerical simulation code developed primarily for simulations of small regions (such as a group of galaxies) in a cosmological context. It integrates the equations of…
During relativistic magnetic reconnection, antiparallel magnetic fields undergo a rapid change in topology, releasing a large amount of energy in the form of non-thermal particle acceleration. This work explores the application of mesh…