arXiv subjects

Xiu Ye

Publications and source records attributed to Xiu Ye.

At least 19 recordsLinked to original sources

arXiv2023

A Modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell's equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

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arXiv2023

Generalized weak Galerkin finite element methods for second order elliptic problems

This article proposes and analyzes the generalized weak Galerkin ({\rm g}WG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the {\rm g}WG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual $L^2$ norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed {\rm g}WG method.

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arXiv2020

A stabilizer free, pressure robust, and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh

In this paper, we propose a new stabilizer free and pressure robust WG method for the Stokes equations with super-convergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal-order for velocity in both energy-norm and the $L^2$-norm and for pressure in $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in the classical discretization. Moreover, the velocity error in our proposed scheme is proved to be independent of pressure and thus confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but remains the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.

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arXiv2020

A $P_{k+2}$ polynomial lifting operator on polygons and polyhedrons

A $P_{k+2}$ polynomial lifting operator is defined on polygons and polyhedrons. It lifts discontinuous polynomials inside the polygon/polyhedron and on the faces to a one-piece $P_{k+2}$ polynomial. With this lifting operator, we prove that the weak Galerkin finite element solution, after this lifting, converges at two orders higher than the optimal order, in both $L^2$ and $H^1$ norms. The theory is confirmed by numerical solutions of 2D and 3D Poisson equations.

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arXiv2020

A stabilizer free WG Method for the Stokes Equations with order two superconvergence on polytopal mesh

A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the $L^2$ norm. Optimal order error estimate for pressure in the $L^2$ norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

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arXiv2020

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

A weak Galerkin (WG) finite element method without stabilizers was introduced in [J. Comput. Appl. Math., 371 (2020). arXiv:1906.06634] on polytopal mesh. Then it was improved in [arXiv:2008.13631] with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the $L^2$ norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.

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arXiv2020

A mixed finite element method on polytopal mesh

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

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arXiv2020

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part II

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (J. Comput. Appl. Math, 371 (2020) 112699. arXiv:1906.06634.) Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem.

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arXiv2020

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.

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arXiv2020

A stabilizer-free pressure-robust finite element method for the Stokes equations

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation. This method employs $H(div)$ finite elements to approximate velocity, which leads to two unique advantages: exact divergence free velocity field and pressure-robustness. In addition, this method has a simple formulation without any stabilizer or penalty term. 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.

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arXiv2020

De Rham Complexes for Weak Galerkin Finite Element Spaces

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element methods on general polyhedral elements. One of the sequences uses polynomials of equal order for all the finite element spaces involved in the sequence and the other one uses polynomials of naturally decending orders. It is shown that the diagrams in both de Rham complexes commute for general polyhedral elements. The exactness of one of the complexes is established for the lowest order element.

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arXiv2020

Numerical investigation on weak Galerkin finite elements

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. The novel idea of weak Galerkin finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces generates different weak Galerkin finite elements. The purpose of this paper is to study stability, convergence and supercloseness of different WG elements by providing many numerical experiments recorded in 31 tables. These tables serve two purposes. First it provides a detail guide of the performance of different WG elements. Second, the information in the tables opens new research territory why some WG elements outperform others.

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arXiv2020

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. The numerical examples are tested on various meshes and confirm the theory.

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arXiv2020

A stabilizer free weak Galerkin element method with supercloseness of order two

The weak Galerkin (WG) finite element method is an effective and flexible general numerical techniques for solving partial differential equations. A simple weak Galerkin finite element method is introduced for second order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory

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arXiv2020

Stabilizer-Free Weak Galerkin Methods for Monotone Quasilinear Elliptic PDEs

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.

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arXiv2019

A conforming DG method for the biharmonic equation on polytopal meshes

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.

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arXiv2019

A stabilizer free weak Galerkin method for the Biharmonic Equation on Polytopal Meshes

A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete $H^2$ for $k\ge 2$ and in $L^2$ norm for $k>2$ are established for the corresponding weak Galerkin finite element solutions. Numerical results are provided to confirm the theories.

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arXiv2019

On stabilizer-free weak Galerkin finite element methods on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous finite element methods will simplify the formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin finite element method for second order elliptic equations on polytopal meshes in 2D or 3D. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyheral meshes. Optimal order error estimates are established for the corresponding WG approximations in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented verifying the theorem.

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