Finite element methods of least-squares type. (English) Zbl 0914.65108

Bochev, Pavel B.; Gunzburger, Max D.

SIAM Rev. 40, No. 4, 789-837 (1998).

Both auhors have many significant contributions on “finite element methods of least squares type”. This paper presents a selective account of past and ongoing work of the developments in least squares finite element methods with a strong focus on the advances for the Stokes and Navier-Stokes equations, for convection-diffusion elliptic problems, inviscid, compressibe flows problems and electromagnetic problems.
The presentation includes general formulation, analysis and implementation of such least squares finite element methods for the different types of problems mentioned above. It is completed by a brief review of other related methods (collocation, restricted least squares and least squares/optimization methods). References to 120 papers are included and commented.
This is a very nice and basic paper to anyone interested on this subject.

Reviewer: M.Bernadou (Le Chesnay)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M10	Finite element methods applied to problems in fluid mechanics
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
35Q30	Navier-Stokes equations
35Q60	PDEs in connection with optics and electromagnetic theory
76D05	Navier-Stokes equations for incompressible viscous fluids
35J25	Boundary value problems for second-order elliptic equations

Keywords:

survey paper; elliptic equations; compressible flows; least-squares finite element methods; Navier-Stokes equations; convection-diffusion elliptic problems; inviscid; electromagnetic problems