Compatible finite element methods for geophysical fluid dynamics

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   Thirty years of the finite volume method for solid mechanics

   Philip Cardiff, Ismet Demirdžić · 2018 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finish…

2. Apparenté Pas pertinent

   On the implementation of the eXtended Finite Element Method (XFEM) for interface problems

   Thomas Carraro, Sven Wetterauer · 2015 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate quadrature rule; (ii) the shape functions for the extended part of the finite element formulation; (iii) the boundary and interface conditions. We show how to handle the XFEM formulation providing a code that demonstrates the solution of two exemplary interface problems for a strong and a weak discontinuity respectively. In the weak discontinuity case, the loss of conformity due to the blending effect and its remedy are discussed. Furthermore, the optimal convergen…

3. Apparenté Pas pertinent

   Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fields

   Dean Muir, Kenneth Duru, Matthew Hole et al. · 2023 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing …

4. Apparenté Pas pertinent

   An overview on deep learning-based approximation methods for partial differential equations

   Christian Beck, Martin Hutzenthaler, Arnulf Jentzen et al. · 2020 · Discrete Contin. Dyn. Syst. Ser. B 28 (2023), no. 6, 3697-3746 · Résumé IA disponible

   Résumé scientifique

   It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ide…

5. Apparenté Pas pertinent

   A robust solution strategy for the Cahn-Larché equations

   Erlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment…

6. Apparenté Pas pertinent

   An embedded corrector problem for homogenization. Part II: Algorithms and discretization

   Eric Cancès, Virginie Ehrlacher, Frederic Legoll et al. · 2018 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   This contribution is the numerically oriented companion article of the work [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Canc\`es, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv p…

7. Apparenté Pas pertinent

   Gradual Variation Analysis for Groundwater Flow

   Li Chen · 2010 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   Groundwater flow in Washington DC greatly influences the surface water quality in urban areas. The current methods of flow estimation, based on Darcy's Law and the groundwater flow equation, can be described by the diffusion equation (the transient flow) and the Laplace equation (the steady-state flow). The Laplace equation is a simplification of the diffusion equation under the condition that the aquifer has a recharging boundary. The practical way of calculation is to use numerical methods to solve these equations. The most popular system is called MODFLOW, which was developed by USGS. MODFLOW is based on the finite-difference method in rectangular Cartesian coordinates. MODFLOW can be vie…

8. Apparenté Pas pertinent

   DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers

   David Widmann, Chris Rackauckas · 2022 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight som…