Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions. (English) Zbl 1442.65380
Summary: 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 [D. Li et al., “Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions”, Preprint, arXiv:1804.03998] from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the \(L^2\)-norm arrives at a superconvergence order of \(\mathcal{O}(h^r)(1.5\leq r\leq 2)\) 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.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
weak Galerkin finite element method; superconvergence; non-uniform cubic partitions; second order elliptic problem; three dimensionsReferences:
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