2 research outputs found
Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions
This article presents a superconvergence for the gradient approximation of
the second order elliptic equation discretized by the weak Galerkin finite
element methods on nonuniform rectangular partitions. The result shows a
convergence of , , for the numerical gradient
obtained from the lowest order weak Galerkin element consisting of piecewise
linear and constant functions. For this numerical scheme, the optimal order of
error estimate is for the gradient approximation. The
superconvergence reveals a superior performance of the weak Galerkin finite
element methods. Some computational results are included to numerically
validate the superconvergence theory
Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions
A superconvergence error estimate for the gradient approximation of the
second order elliptic problem in three dimensions is analyzed by using weak
Galerkin finite element scheme on the uniform and non-uniform cubic partitions.
Due to the loss of the symmetric property from two dimensions to three
dimensions, this superconvergence result in three dimensions is not a trivial
extension of the recent superconvergence result in two dimensions
\cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error
estimate for the numerical gradient in the -norm arrives at a
superconvergence order of when the lowest
order weak Galerkin finite elements consisting of piecewise linear polynomials
in the interior of the elements and piecewise constants on the faces of the
elements are employed. A series of numerical experiments are illustrated to
confirm the established superconvergence theory in three dimensions.Comment: 31 pages, 24 table