Publication

Consistency, Stability, a-Priori and a-Posteriori Errors for Petrov-Galerkin Methods Applied to Nonlinear Problems

Jacques Rappaz
1994
Journal Articles

Abstract

In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: -div (k(u)DELTAu) + c. DELTAu = f in a two dimensional domain OMEGA with Dirichlet boundary conditions.

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https://infoscience.epfl.ch/entities/publication/07362ec7-5f75-448c-a17f-30f1205151f9

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Jacques Rappaz
1994
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/07362ec7-5f75-448c-a17f-30f1205151f9

About this result

Related concepts (32)

Boundary value problem

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Related publications (21)

Consistency, Stability, a-Priori and a-Posteriori Errors for Petrov-Galerkin Methods Applied to Nonlinear Problems

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