Browsing by Author "Ye, Xiu"
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Item A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids(Electronic Research Archive, 2024-05-27) Yang, Yan; Ye, Xiu; Zhang, ShangyouA pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the L2 norm and the H1 norm. The optimal-order error convergence has been proved for the pressure in the L2 norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.Item A superconvergent CDG finite element for the Poisson equation on polytopal meshes(Zeitschrift für anorganische und allgemeine Chemie | Journal of Inorganic and General Chemistry, 2023-12-08) Ye, Xiu; Zhang, ShangyouA conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted Pk+2 function from four Pk functions. When the numerical gradient space is the H (div,T) subspace of piecewise Pdk+1 polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T which have a one-piece polynomial divergence on T, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post-process which lifts a Pk CDG solution to a quasi-optimal Pk+2 solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.Item Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: The CDG Method(East Asian Journal on Applied Mathematics, 2022-04-06) Ye, Xiu; Zhang, ShangyouNovelty of this work is the development of a finite element method using discontinuous Pk element, which has two-order higher convergence rate than the optimal order. The method is used to solve a one-dimensional second order elliptic problem. A totally new approach is developed for error analysis. Superconvergence of order two for the CDG finite element solution is obtained. The Pk solution is lifted to an optimal order Pk+2 solution elementwise. The numerical results confirm the theory.Item Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes(CSIAM Transactions on Applied Mathematics, 2023-02) Ye, Xiu; Zhang, ShangyouIt is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous Pk elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous Pk element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L2 norm. A local post-process is defined which lifts a Pk CDG solution to a discontinuous Pk+2 solution. It is proved that the lifted Pk+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the Pk CDG finite element is as good as the Pk+2 continuous Galerkin finite element.Item A stabilizer-free pressure-robust finite element method for the Stokes equations(Advances in Computational Mathematics, 2021-04-08) Ye, Xiu; Zhang, ShangyouIn this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.Item Two-order superconvergence for a weak Galerkin method on rectangular and cuboid grids(Numerical Methods for Partial Differential Equations, 2022-09-22) Wang, Junping; Wang, Xiaoshen; Ye, Xiu; Zhang, Shangyou; Zhu, PengThis article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses k $$ k $$ th order polynomial for weak finite element functions and ( k + 1 ) $$ \left(k+1\right) $$ th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the L 2 $$ {L}^2 $$ norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element.