The Meshless Local Petrov-Galerkin (MLPG) Method for Solving Incompressible Navier-Stokes Equations
H. Lin and S.N. Atluri

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 2, No. 2, pp. 117-142, 2001
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Keywords MLPG, MLS, Babu\~{s}ka-Brezzi conditions, upwinding scheme, incompressible flow, Navier-Stokes equations.
Abstract The truly Meshless Local Petrov-Galerkin (MLPG) method is extended to solve the incompressible Navier-Stokes equations. The local weak form is modified in a very careful way so as to ovecome the so-called Babu\~{s}ka-Brezzi conditions. In addition, The upwinding scheme as developed in \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{Lin2000} and \relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{Lin2000a} is used to stabilize the convection operator in the streamline direction. Numerical results for benchmark problems show that the MLPG method is very promising to solve the convection dominated fluid mechanics problems.
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