Volume 4, Issue 3-4
Analysis of the [L2, L2, L2] Least Squares Finite Element Method for Incompressible Oseen-type Problems

C. L. Chang & S.-Y. Yang

DOI:

Int. J. Numer. Anal. Mod., 4 (2007), pp. 402-424.

Published online: 2007-04

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  • Abstract

In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [L-2, L-2, L-2] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L-2 norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the H-1 x L-2 x L-2 norm for the velocity, vorticity, and pressure, but only continuous in the H-1 x H-1 x H-1 norm for these variables. although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H-1 norm for velocity and suboptimal rate of convergence in the L-2 norm for vorticity and pressure. numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L-2, L-2, L-2] least-squares finite element approach.

  • Keywords

Navier-Stokes equations Oseen-type equations finite element methods least squares

  • AMS Subject Headings

65N15 65N30 76M10

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-402, author = {}, title = {Analysis of the [L2, L2, L2] Least Squares Finite Element Method for Incompressible Oseen-type Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {3-4}, pages = {402--424}, abstract = {In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [L-2, L-2, L-2] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L-2 norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the H-1 x L-2 x L-2 norm for the velocity, vorticity, and pressure, but only continuous in the H-1 x H-1 x H-1 norm for these variables. although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H-1 norm for velocity and suboptimal rate of convergence in the L-2 norm for vorticity and pressure. numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L-2, L-2, L-2] least-squares finite element approach. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/869.html} }
TY - JOUR T1 - Analysis of the [L2, L2, L2] Least Squares Finite Element Method for Incompressible Oseen-type Problems JO - International Journal of Numerical Analysis and Modeling VL - 3-4 SP - 402 EP - 424 PY - 2007 DA - 2007/04 SN - 4 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/869.html KW - Navier-Stokes equations KW - Oseen-type equations KW - finite element methods KW - least squares AB - In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [L-2, L-2, L-2] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L-2 norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the H-1 x L-2 x L-2 norm for the velocity, vorticity, and pressure, but only continuous in the H-1 x H-1 x H-1 norm for these variables. although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H-1 norm for velocity and suboptimal rate of convergence in the L-2 norm for vorticity and pressure. numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L-2, L-2, L-2] least-squares finite element approach.
C. L. Chang & S.-Y. Yang. (2019). Analysis of the [L2, L2, L2] Least Squares Finite Element Method for Incompressible Oseen-type Problems. International Journal of Numerical Analysis and Modeling. 4 (3-4). 402-424. doi:
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