TY - JOUR
T1 - First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems
AU - Chen, Huangxin
AU - Fu, Guosheng
AU - Li, Jingzhi
AU - Qiu, Weifeng
PY - 2014/12
Y1 - 2014/12
N2 - We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data fεL2(Ω). The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov-Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by Gopalakrishnan and Qiu (2014). This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform.
AB - We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data fεL2(Ω). The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov-Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by Gopalakrishnan and Qiu (2014). This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform.
KW - Convection diffusion problems
KW - DPG method
KW - Error estimate
KW - Least squares method
UR - http://www.scopus.com/inward/record.url?scp=84919680128&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84919680128&origin=recordpage
U2 - 10.1016/j.camwa.2014.11.001
DO - 10.1016/j.camwa.2014.11.001
M3 - 21_Publication in refereed journal
VL - 68
SP - 1635
EP - 1652
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
SN - 0898-1221
IS - 12
ER -