arXiv subjects

Shangyou Zhang

Publications and source records attributed to Shangyou Zhang.

At least 19 recordsLinked to original sources

arXiv2026

A Least-Squares Weak Galerkin Method for the Biharmonic Cauchy Problem

We develop a least-squares weak Galerkin (LS-WG) finite element method for the Cauchy problem of the biharmonic equation. The proposed approach reformulates the fourth-order equation as a coupled system of two second-order equations, which are discretized using discrete weak Laplacian operators on weak finite element spaces. The resulting least-squares formulation yields a symmetric positive definite linear system, thereby eliminating the discrete inf-sup condition required by mixed finite element methods while avoiding the construction of globally $C^1$-conforming finite element spaces. Furthermore, the weak Galerkin framework naturally accommodates general polygonal meshes, offering considerable flexibility in mesh generation and approximation. Under the assumption that the continuous biharmonic Cauchy problem admits a unique solution, we establish the uniqueness of the discrete LS-WG solution and derive optimal-order error estimates in a discrete energy norm. Numerical experiments confirm the theoretical convergence rates and demonstrate the accuracy, robustness, and effectiveness of the proposed method.

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arXiv2026

A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes

This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method's stability, efficiency, and locking-free performance for nearly incompressible materials.

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arXiv2026

A Least Squares Weak Galerkin Finite Element Method for Fokker-Planck Type Equations

This paper presents a least squares weak Galerkin (LS-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. To address the numerical challenges arising from non-smooth diffusion tensors, the proposed method utilizes a least-squares formulation that yields a symmetric positive definite (SPD) discrete system. The numerical scheme is designed by employing locally constructed weak second order partial derivatives and the weak divergence commonly used within the weak Galerkin framework. A rigorous theoretical foundation is provided, establishing the uniqueness of the discrete solution and deriving optimal-order error estimates in a discrete energy norm. Finally, extensive numerical experiments are reported to validate the theoretical findings and demonstrate the robustness and performance of the numerical scheme.

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arXiv2026

Solving the Stokes Equations via a Least Squares Weak Galerkin Method

We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of $\mathcal{O}(h^k)$ for the discrete projection error and $\mathcal{O}(h^{k-1})$ for the global approximation error when employing polynomials of degree $k \ge 1$ for the velocity field and $k-1$ for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method's robustness, geometric flexibility, and overall efficiency.

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arXiv2026

A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz

This paper introduces and rigorously analyzes a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation. By utilizing a weak Laplacian operator defined on a space of discontinuous functions, the proposed framework facilitates the seamless treatment of complex boundary conditions and internal interfaces. We emphasize the geometric flexibility of the LS-WG scheme on general polygonal and polyhedral partitions. Furthermore, we prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm. Extensive numerical experiments validate the theoretical convergence rates and demonstrate the algorithm's robustness and efficiency over traditional Galerkin approaches.

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arXiv2026

A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection--Diffusion

We introduce and rigorously analyze a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem of convection--diffusion equations. The proposed framework utilizes weak derivatives defined on a class of discontinuous weak functions, enabling the natural treatment of complex boundary conditions and internal interfaces. A key advantage of the least-squares formulation is that it transforms the underlying non-self-adjoint operator into a discrete linear system that is inherently symmetric and positive definite (SPD). We demonstrate the geometric flexibility of the method on arbitrary polygonal and polyhedral partitions. Furthermore, we establish the uniqueness of the numerical solution and derive optimal-order error estimates in a carefully defined discrete energy norm. Extensive numerical tests are presented to confirm the theoretical convergence rates and highlight the algorithm's robustness and efficiency compared to standard Galerkin approaches.

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arXiv2026

A Least-Squares Weak Galerkin Method for Second-Order Elliptic Equations in Non-Divergence Form

This article proposes a novel least-squares weak Galerkin (LS-WG) method for second-order elliptic equations in non-divergence form. The approach leverages a locally defined discrete weak Hessian operator constructed within the weak Galerkin framework. A key feature of the resulting algorithm is that it yields a symmetric and positive definite linear system while remaining applicable to general polygonal and polyhedral meshes. We establish optimal-order error estimates for the approximation in a discrete $H^2$-equivalent norm. Finally, comprehensive numerical experiments are presented to validate the theoretical analysis and demonstrate the efficiency and robustness of the method.

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arXiv2026

Auto-Stabilized Weak Galerkin Finite Element Methods for Biot's consolidation model on Non-Convex Polytopal Meshes

This paper presents an auto-stabilized weak Galerkin (WG) finite element method for the Biot's consolidation model within the classical displacement-pressure two-field formulation. Unlike traditional WG approaches, the proposed scheme achieves numerical stability without the requirement of traditional stabilizers. Spatial discretization is performed using weak Galerkin finite elements for both displacement and pressure approximations, while a backward Euler scheme is employed for temporal discretization to ensure a fully implicit and stable formulation. We establish the well-posedness of the resulting linear system at each time step and provide a rigorous error analysis, deriving optimal-order convergence. A significant merit of this WG scheme is its flexibility on general shape-regular polytopal meshes, including those with non-convex geometries. By utilizing bubble functions as a primary analytical tool, the method produces stable, oscillation-free pressure approximations without specialized treatment. Numerical experiments are presented to validate the theoretical convergence rates and demonstrate the computational efficiency and robustness of the auto-stabilized formulation.

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arXiv2026

A Simple Weak Galerkin Finite Element Method for Convection-Diffusion-Reaction Equations on Nonconvex Polytopal Meshes

This article introduces a simple weak Galerkin (WG) finite element method for solving convection-diffusion-reaction equation. The proposed method offers significant flexibility by supporting discontinuous approximating functions on general nonconvex polytopal meshes. We establish rigorous error estimates within a suitable norm. Finally, numerical experiments are presented to validate the theoretical convergence rates and demonstrate the computational efficiency of the approach.

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arXiv2026

A weak Galerkin least squares finite element method for linear convection equations in non-divergence form

This article develops a weak Galerkin least-squares (WG--LS) finite element method for first-order linear convection equations in non-divergence form. The method is formulated using discontinuous finite element functions and does not require any coercivity assumption on the convection vector or reaction coefficient. The resulting discrete problem leads to a symmetric and positive definite linear system and is applicable to general polygonal and polyhedral meshes. Under minimal regularity assumptions on the coefficients, optimal-order error estimates are established for the WG--LS approximation in a suitable energy norm. Numerical experiments are presented to confirm the theoretical convergence results and to demonstrate the accuracy and efficiency of the proposed method.

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arXiv2025

Weak Galerkin finite element methods for elliptic interface problems on nonconvex polygonal partitions

This paper proposes a weak Galerkin (WG) finite element method for elliptic interface problems defined on nonconvex polygonal partitions. The method features a built-in stabilizer and retains a simple, symmetric, and positive definite formulation. An optimal-order error estimate is rigorously derived in the discrete $H^1$ norm. Furthermore, a series of numerical experiments are provided to verify the theoretical results and to demonstrate the robustness and effectiveness of the proposed WG method for elliptic interface problems.

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arXiv2025

$C^1$-$Q_k$ serendipity finite elements on rectangular meshes

A $C^1$-$Q_k$ serendipity finite element is a sub-element of $C^1$-$Q_k$ BFS finite element such that the element remains $C^1$-continuous and includes all $P_k$ polynomials. In other words, it is a minimum of $Q_k$ bubbles enriched $P_k$ finite element. We enrich the $P_4$ and $P_5$ spaces by $9$ $Q_4$ and $11$ $Q_5$-bubble functions, respectively. For all $k\ge 6$, we enrich the $P_k$ spaces exactly by $12$ $Q_k$ bubble functions. We show the uni-solvence and quasi-optimality of the newly defined $C^1$-$Q_k$ serendipity elements. Numerical experiments by the $C^1$-$Q_k$ serendipity elements, $4\le k\le 8$, are performed.

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arXiv2025

Rectangular $C^1$-$P_k$ finite elements with $Q_k$-bubble enrichment

We enrich the $P_k$ polynomial space by $5$ ($k=4$), or $7$ ($k=5$), or 8 (all $k\ge 6$) $Q_k$ bubble functions to obtain a family of $C^1$-$P_k$ ($k\ge 4$) finite elements on rectangular meshes. We show the uni-solvency, the $C^1$-continuity and the quasi-optimal convergence. Numerical tests on the new $C^1$-$P_k$, $k=4,5,6,7$ and $8$, elements are performed.

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arXiv2025

A Simple Weak Galerkin Finite Element Method for the Reissner-Mindlin Plate Model on Non-Convex Polytopal Meshes

This paper presents a simple weak Galerkin (WG) finite element method for the Reissner-Mindlin plate model that partially eliminates the need for traditionally employed stabilizers. The proposed approach accommodates general, including non-convex, polytopal meshes, thereby offering greater geometric flexibility. It utilizes bubble functions without imposing the restrictive conditions required by existing stabilizer-free WG methods, which simplifies implementation and broadens applicability to a wide range of partial differential equations (PDEs). Moreover, the method allows for flexible choices of polynomial degrees in the discretization and can be applied in any spatial dimension. We establish optimal-order error estimates for the WG approximation in a discrete H^1 norm, and present numerical experiments that validate the theoretical results.

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arXiv2025

A Simple and Robust Weak Galerkin Method for the Brinkman Equations on Non-Convex Polytopal Meshes

This paper presents a novel Stabilizer-Free weak Galerkin (WG) finite element method for solving the Brinkman equations without the need for conventional stabilization techniques. The Brinkman model, which mathematically blends features of both the Stokes and Darcy equations, describes fluid flow in multi-physics environments, particularly in heterogeneous porous media characterized by spatially varying permeability. In such settings, flow behavior may be governed predominantly by Darcy dynamics in certain regions and by Stokes dynamics in others. A central difficulty in this context arises from the incompatibility of standard finite element spaces: elements stable for the Stokes equations typically perform poorly for Darcy flows, and vice versa. The primary challenge addressed in this study is the development of a unified numerical scheme that maintains stability and accuracy across both flow regimes. To this end, the proposed WG method demonstrates a robust capacity to resolve both Stokes- and Darcy-dominated flows through a unified framework. The method supports general finite element partitions consisting of convex and non-convex polytopal elements, and employs bubble functions as a critical analytical component to achieve stability and convergence. Optimal-order error estimates are rigorously derived for the WG finite element solutions. Additionally, a series of numerical experiments is conducted to validate the theoretical findings, illustrating the method's robustness, reliability, flexibility, and accuracy in solving the Brinkman equations.

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arXiv2025

Weak Galerkin Methods for the Brinkman Equations

This paper introduces a novel weak Galerkin (WG) finite element method for the numerical solution of the Brinkman equations. The Brinkman model, which seamlessly integrates characteristics of both the Stokes and Darcy equations, is employed to describe fluid flow in multiphysics contexts, particularly within heterogeneous porous media exhibiting spatially variable permeability. The proposed WG method offers a unified and robust approach capable of accurately capturing both Stokes- and Darcy-dominated regimes. A discrete inf-sup condition is established, and optimal-order error estimates are rigorously proven for the WG finite element solutions. Furthermore, a series of numerical experiments is performed to corroborate the theoretical analysis, demonstrating the method's accuracy and stability in addressing the complexities inherent in the Brinkman equations.

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arXiv2025

Rectangular $C^1$-$Q_k$ Bell finite elements in two and three dimensions

Both the function and its normal derivative on the element boundary are $Q_k$ polynomials for the Bogner-Fox-Schmit $C^1$-$Q_k$ finite element functions. Mathematically, to keep the optimal order of approximation, their spaces are required to include $P_k$ and $P_{k-1}$ polynomials respectively. We construct a Bell type $C^1$-$Q_k$ finite element on rectangular meshes in 2D and 3D, which has its normal derivative as a $Q_{k-1}$ polynomial on each face, for $k\ge 4$. We show, with a big reduction of the space, the $C^1$-$Q_k$ Bell finite element retains the optimal order of convergence. Numerical experiments are performed, comparing the new elements with the original elements.

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arXiv2025

A nodal basis for the $C^1$-$P_{33}$ finite elements on 5D simplex grids

We construct a nodal basis for the 5-dimensional $C^1$ finite element space of polynomial degree $33$ on simplex grids, where the finite element functions are $C^1$ on the 6 4D-simplex faces, $C^2$ on the 15 face-tetrahedra, $C^4$ on the 20 face-triangles, $C^8$ on the 15 edges, and $C^{16}$ at the 6 vertices, of a 5D simplex.

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