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 - <p style="text-align: justify;">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.</p>