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Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in ℝ N , u ∈ H 1 ( ℝ N ) , where N ≥ 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N ≥ 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N ≥ 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k ⊂ O ( 2 ) denotes the dihedral group...

First kind integral equations for the numerical solution of the plane Dirichlet problem

Søren Christiansen (1989)

Aplikace matematiky

We present, in a uniform manner, several integral equations of the first kind for the solution of the two-dimensional interior Dirichlet boundary value problem. We apply a general numerical collocation method to the various equations, and thereby we compare the various integral equations, and recommend two of them. We give a survey of the various numerical methods, and present a simple method for the numerical solution of the recommended integral equations.

Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

Paola Causin, Riccardo Sacco, Carlo L. Bottasso (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum...

Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

Paola Causin, Riccardo Sacco, Carlo L. Bottasso (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete...

Fokker-Planck equation in bounded domain

Laurent Chupin (2010)

Annales de l’institut Fourier

We study the existence and the uniqueness of a solution  ϕ to the linear Fokker-Planck equation - Δ ϕ + div ( ϕ F ) = f in a bounded domain of  ℝ d when F is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.

Formulations Mixtes Augmentées et Applications

Boujemâa Achchab, Abdellatif AGOUZAL (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and analyse a abstract framework for augmented mixed formulations. We give a priori error estimate in the general case: conforming and nonconforming approximations with or without numerical integration. Finally, a posteriori error estimator is given. An example of stabilized formulation for Stokes problem is analysed.

Generalizations of the Finite Element Method

Marc Schweitzer (2012)

Open Mathematics

This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To...

Generic implementation of finite element methods in the Distributed and Unified Numerics Environment (DUNE)

Peter Bastian, Felix Heimann, Sven Marnach (2010)

Kybernetika

In this paper we describe PDELab, an extensible C++ template library for finite element methods based on the Distributed and Unified Numerics Environment (Dune). PDELab considerably simplifies the implementation of discretization schemes for systems of partial differential equations by setting up global functions and operators from a simple element-local description. A general concept for incorporation of constraints eases the implementation of essential boundary conditions, hanging nodes and varying...

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