arXiv subjects

Shangyou Zhang

Publications and source records attributed to Shangyou Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Superconvergent quadriatic finite element on uniform tetrahedral meshes

By a direct computation, we show that the $P_2$ interpolation of a $P_3$ function is also a local $H^1$-projection on uniform tetrahedral meshes, i.e., the difference is $H^1$-orthogonal to the $P_2$ Lagrange basis function on the support patch of tetrahedra of the basis function. Consequently, we show the $H^1$ and $L^2$ rconvergence of the $P_2$ Lagrange finite element on uniform tetrahedral meshes. Using the standard 20-points Lagrange $P_3$ interpolation, where the 20 nodes are exactly some $P_2$ global basis nodes, we lift the superconvergent $P_2$ finite element solution to a quasi-optimal $P_3$ solution on each cube. Numerical results confirm the theory.

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arXiv2025

Two families of C1-Pk Fraeijs de Veubeke-Sander finite elements on quadrilateral meshes

We extend the $C^1$-$P_3$ Fraeijs de Veubeke-Sander finite element to two families of $C^1$-$P_k$ ($k>3$) macro finite elements on general quadrilateral meshes. On each quadrilateral, four $P_k$ polynomials are defined on the four triangles subdivided from the quadrilateral by its two diagonal lines. The first family of $C^1$-$P_k$ finite elements is the full $C^1$-$P_k$ space on the macro-mesh. Thus the element can be applied to interface problems. The second family of $C^1$-$P_k$ finite elements condenses all internal degrees of freedom by moving them to the four edges. Thus the second element method has much less unknowns but is more accurate than the first one. We prove the uni-solvency and the optimal order convergence. Numerical tests and comparisons with the $C^1$-$P_k$ Argyris are provided.

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arXiv2025

A Simple Weak Galerkin Finite Element Method for a Class of Fourth-Order Problems in Fluorescence Tomography

In this paper, we propose a simple numerical algorithm based on the weak Galerkin (WG) finite element method for a class of fourth-order problems in fluorescence tomography (FT), eliminating the need for stabilizer terms required in traditional WG methods. FT is an emerging, non-invasive 3D imaging technique that reconstructs images of fluorophore-tagged molecule distributions in vivo. By leveraging bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, representing a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches, offering substantial advantages. The proposed method preserves a simple, symmetric, and positive definite structure. These advantages are confirmed by optimal-order error estimates in a discrete $H^2$ norm, demonstrating the effectiveness and accuracy of our approach. Numerical experiments further validate the efficiency and precision of the proposed method.

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arXiv2025

Stabilizer-free Weak Galerkin Methods for Quad-Curl Problems on polyhedral Meshes without Convexity Assumptions

This paper introduces an efficient stabilizer-free weak Galerkin (WG) finite element method for solving the three-dimensional quad-curl problem. Leveraging bubble functions as a key analytical tool, the method extends the applicability of stabilizer-free WG approaches to non-convex elements in finite element partitions-a notable advancement over existing methods, which are restricted to convex elements. The proposed method maintains a simple, symmetric, and positive definite formulation. It achieves optimal error estimates for the exact solution in a discrete norm, as well as an optimal-order $L^2$ error estimate for $k>2$ and a sub-optimal order for the lowest order case $k=2$. Numerical experiments are presented to validate the method's efficiency and accuracy.

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arXiv2025

An Auto-Stabilized Weak Galerkin Method for Elasticity Interface Problems on Nonconvex Meshes

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for elasticity interface problems on general polygonal and polyhedral meshes, without requiring convexity constraints. The method utilizes bubble functions as key analytical tools, eliminating the need for stabilizers typically used in traditional WG methods and leading to a more streamlined formulation. The proposed method is symmetric, positive definite, and easy to implement. Optimal-order error estimates are derived for the WG approximations in the discrete $H^1$-norm, assuming the exact solution has sufficient smoothness. Numerical experiments validate the accuracy and efficiency of the auto-stabilized WG method.

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arXiv2025

Auto-Stabilized Weak Galerkin Finite Element Methods for Stokes Equations on Non-Convex Polytopal Meshes

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for solving Stokes equations without relying on traditional stabilizers. The proposed WG method accommodates both convex and non-convex polytopal elements in finite element partitions, leveraging bubble functions as a key analytical tool. The simplified WG method is symmetric and positive definite, and optimal-order error estimates are derived for WG approximations in both the discrete $H^1$ norm and the $L^2$ norm.

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arXiv2024

Simplified Weak Galerkin Methods for Linear Elasticity on Nonconvex Domains

This paper presents a weak Galerkin (WG) finite element method for linear elasticity on general polygonal and polyhedral meshes, free from convexity constraints, by leveraging bubble functions as central analytical tools. The proposed method eliminates the need for stabilizers commonly used in traditional WG methods, resulting in a simplified formulation. The method is symmetric, positive definite, and straightforward to implement. Optimal-order error estimates are established for the WG approximations in the discrete $H^1$-norm, assuming sufficient smoothness of the exact solution, and in the standard $L^2$-norm under regularity assumptions for the dual problem. Numerical experiments confirm the efficiency and accuracy of the proposed stabilizer-free WG method.

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arXiv2024

Efficient Weak Galerkin Finite Element Methods for Maxwell Equations on polyhedral Meshes without Convexity Constraints

This paper presents an efficient weak Galerkin (WG) finite element method with reduced stabilizers for solving the time-harmonic Maxwell equations on both convex and non-convex polyhedral meshes. By employing bubble functions as a critical analytical tool, the proposed method enhances efficiency by partially eliminating the stabilizers traditionally used in WG methods. This streamlined WG method demonstrates stability and effectiveness on convex and non-convex polyhedral meshes, representing a significant improvement over existing stabilizer-free WG methods, which are typically limited to convex elements within finite element partitions. The method achieves an optimal error estimate for the exact solution in a discrete $H^1$ norm, and additionally, an optimal $L^2$ error estimate is established for the WG solution. Several numerical experiments are conducted to validate the method's efficiency and accuracy.

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arXiv2024

A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids

A nonconforming $P_3$ finite element is constructed by enriching the conforming $P_3$ finite element space with three $P_3$ nonconforming bubbles and six additional $P_4$ nonconforming bubbles, on each tetrahedron. Here the divergence of the $P_4$ bubble is not a $P_3$ polynomial, but a $P_2$ polynomial. This nonconforming $P_3$ finite element, combined with the discontinuous $P_2$ finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special $P_4$ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.

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arXiv2024

A nonconforming P2 and discontinuous P1 mixed finite element on tetrahedral grids

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergen solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

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arXiv2024

Convergent finite elements on arbitrary meshes, the WG method

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e., on arbitrary meshes. Numerical tests confirm the theory.

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arXiv2024

New Analysis of Overlapping Schwarz Methods for Vector Field Problems in Three Dimensions with Generally Shaped Domains

This paper introduces a novel approach to analyzing overlapping Schwarz methods for Nédélec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.

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arXiv2023

A family of stabilizer-free virtual elements on triangular meshes

A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The interpolating space here is the space of continuous $P_k$ polynomials on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined by connecting three vertices of a triangle with its barycenter. We show that such an interpolation preserves $P_k$ polynomials locally and enforces the coerciveness of the resulting bilinear form. Consequently the stabilizer-free virtual element solutions converge at the optimal order. Numerical tests are provided to confirm the theory and to be compared with existing virtual elements.

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arXiv2023

Superconvergent P1 honeycomb virtual elements and lifted P3 solutions

When solving the Poisson equation on honeycomb hexagonal grids, we show that the $P_1$ virtual element is three-order superconvergent in $H^1$-norm, and two-order superconvergent in $L^2$ and $L^\infty$ norms. We define a local post-process which lifts the superconvergent $P_1$ solution to a $P_3$ solution of the optimal-order approximation. The theory is confirmed by a numerical test.

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arXiv2023

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete $H^1$ and $L^2$ norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

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arXiv2023

Weak Galerkin Finite Element Methods for Optimal Control Problems Governed by Second Order Elliptic Partial Differential Equations

This paper is concerned with the development of weak Galerkin (WG) finite element method for optimal control problems governed by second order elliptic partial differential equations (PDEs). It is advantageous to use discontinuous finite elements over the traditional $C^1$ finite elements here. Optimal order error estimates are established and confirmed by some numerical tests.

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arXiv2023

Stabilizer-free polygonal and polyhedral virtual elements

Stabilizer-free $P_k$ virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous $P_k$ polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision of each polyhedron. With such an accurate and proper interpolation, the stabilizer of the virtual elements is eliminated while the system is kept positive-definite. We show that the stabilizer-free virtual elements converge at the optimal order in 2D and 3D. Numerical examples are computed, validating the theory.

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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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