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 Li et al. (0000) from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L 2 -norm arrives at a superconvergence order of O ( h r ) ( 1 . 5 ≤ r ≤ 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.

pdf/superconvergence-of-numerical-gradient-for-weak-galerkin-3syxgldk4a.pdf

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Supraconvergence and supercloseness in quasilinear coupled problems

In this article, we study the semidiscrete H 1-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L 2-norm for the flux is of order &#x1d4aa;(h k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order &#x1d4aa;(h k+3) between the H 1-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order &#x1d4aa;(h k+3) is realized via a postprocessing technique using local projection methods.

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

We derive superconvergence result for H 1-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Babŭska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order &#x1d4aa;(h k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We establish that the approximate solution of the mixed finite element method for the primary function is superconvergent to its $$L_2$$
 local projection on unstructured meshes. To the best of our knowledge, all of the superconvergent results concerning mixed method need some conditions such as requiring the underlying meshes to be structured or using higher-order approximation spaces. Our results are valid on shape regular meshes, and any order of approximation spaces can be used. As an elementary consequence, we provide a proof that solutions of least-squares (LS) finite element method are higher-order perturbations of the solutions of the mixed and standard Galerkin methods. Also, we provide an $$L_2$$
 norm error estimate for LS finite element solution on graded meshes.

Supercloseness of the mixed finite element method for the primary function on unstructured meshes and its applications

A new superconvergence result is established for numerical solutions of elliptic problems obtained from the mixed finite element method of Raviart--Thomas over rectangular partitions. The well-known optimal order error estimate in L2-norm for the flux approximation is of order ${\cal O}(h^{k+1})$, where $k\ge 0$ is the order of polynomials employed in the Raviart--Thomas element. The new superconvergence shows an improved accuracy of order ${\cal O}(h^{k+3})$ between the mixed finite element approximation and an appropriately defined local projection of the flux variable when k>0. A postprocessing technique using local projection methods is proposed and analyzed in order to provide a new approximate solution with the superconvergent order ${\cal O}(h^{k+3})$.

A New Superconvergence for Mixed Finite Element Approximations

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A New Superconvergence for Mixed Finite Element Approximations