Volume 41, Issue 1
Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

J. Comp. Math., 41 (2023), pp. 39-71.

Published online: 2022-11

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• Abstract

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.  We derive error estimates for all unknowns under both $L^2$ norms and energy norms.  Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

• AMS Subject Headings

65N30

rli@math.pku.edu.cn (Ruo Li)

yangfanyi@scu.edu.cn (Fanyi Yang)

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@Article{JCM-41-39, author = {Li , Ruo and Yang , Fanyi}, title = {Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {1}, pages = {39--71}, abstract = {

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.  We derive error estimates for all unknowns under both $L^2$ norms and energy norms.  Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2104-m2020-0231}, url = {http://global-sci.org/intro/article_detail/jcm/21169.html} }
TY - JOUR T1 - Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation AU - Li , Ruo AU - Yang , Fanyi JO - Journal of Computational Mathematics VL - 1 SP - 39 EP - 71 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2104-m2020-0231 UR - https://global-sci.org/intro/article_detail/jcm/21169.html KW - Stokes problem, Least squares finite element method, Reconstructed discontinuous approximation, Solenoid and irrotational polynomial bases. AB -

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.  We derive error estimates for all unknowns under both $L^2$ norms and energy norms.  Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

Ruo Li & Fanyi Yang. (2022). Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation. Journal of Computational Mathematics. 41 (1). 39-71. doi:10.4208/jcm.2104-m2020-0231
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