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Volume 14, Issue 4
A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

Ruo Li & Fanyi Yang

DOI: 10.4208/nmtma.OA-2021-0042

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 1042-1067.

Published online: 2021-09

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  • Abstract

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

  • Keywords

Non-divergence form, least squares method, piecewise irrotational space, discontinuous Galerkin method.

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-1042, author = {Li , Ruo and Yang , Fanyi}, title = {A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {4}, pages = {1042--1067}, abstract = {

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0042 }, url = {http://global-sci.org/intro/article_detail/nmtma/19529.html} }
TY - JOUR T1 - A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form AU - Li , Ruo AU - Yang , Fanyi JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1042 EP - 1067 PY - 2021 DA - 2021/09 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2021-0042 UR - https://global-sci.org/intro/article_detail/nmtma/19529.html KW - Non-divergence form, least squares method, piecewise irrotational space, discontinuous Galerkin method. AB -

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

Li , Ruo and Yang , Fanyi. (2021). A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form. Numerical Mathematics: Theory, Methods and Applications. 14 (4). 1042-1067. doi:10.4208/nmtma.OA-2021-0042
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