The finite element method with Lagrangian multipliers

Babuška, Ivo

doi:10.1007/bf01436561

Cited by 1,447 publications

(1,055 citation statements)

References 8 publications

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“…It is well known that FE spaces V h and Q h for the velocity and the pressure, respectively, cannot be chosen arbitrary if the standard Galerkin method is applied to modelling incompressible flows. They have to meet the so-called Babus ska-Brezzi-Ladyzhenskaya condition (BBL) (Babus ska, 1973;Brezzi, 1974;Ladyzhenskaya and Solonnikov, 1976). Piecewise linear basis functions for the velocity and pressure do not meet this requirement.…”

Section: Stabilization For the Dynamical Partmentioning

confidence: 99%

A finite-element ocean model: principles and evaluation

2004

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We describe a three-dimensional (3D) finite-element ocean model designed for investigating the largescale ocean circulation on time scales from years to decades. The model solves the primitive equations in the dynamical part and the advection-diffusion equations for temperature and salinity in the thermodynamical part. The time-stepping is implicit. The 3D mesh is composed of tetrahedra and has a variable resolution. It is based on an unstructured 2D surface mesh and is stratified in the vertical direction. The model uses linear functions for horizontal velocity and tracers on tetrahedra, and for surface elevation on surface triangles. The vertical velocity field is elementwise constant. An important ingredient of the model is the Galerkin least-squares stabilization used to minimize effects of unresolved boundary layers and make the matrices to be inverted in time-stepping better conditioned. The model performance was tested in a 16-year simulation of the North Atlantic using a mesh covering the area between 7°and 80°N and providing variable horizontal resolution from 0.3°to 1.5°.

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Section: Stabilization For the Dynamical Partmentioning

confidence: 99%

A finite-element ocean model: principles and evaluation

2004

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“…[4,27,110]. The resulting saddle-point problems are solved using mixed finite-element methods [26,9,58,4,27], in which the basis functions ψ j ∈ Ψ Δ and ϕ j ∈ Φ Δ are drawn from different but compatible discrete function spaces, Ψ Δ and Φ Δ . After the assembly of the finite-elements, one ends with a nonlinear sparse system of algebraic equations, A(W Δ ) = G Δ .…”

Section: Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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“…The classical theory requires B to satisfy an infsup condition [8,13,14]. The development of discrete spaces which inherit such inf-sup conditions for each specific problem is notoriously difficult, and this difficulty is compounded by the product structure of the twofold saddle point problems.…”

Section: Problems Of Type (12) Can Be Written Asmentioning

confidence: 99%

Inf–sup conditions for twofold saddle point problems

Howell

Walkington

2011

Numer. Math.

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Summary Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.

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The finite element method with Lagrangian multipliers

Cited by 1,447 publications

References 8 publications

A finite-element ocean model: principles and evaluation

A finite-element ocean model: principles and evaluation

A review of numerical methods for nonlinear partial differential equations

Inf–sup conditions for twofold saddle point problems

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