Article
Engineering, Multidisciplinary
Habeun Choi, Heng Chi, Kyoungsoo Park, Glaucio H. Paulino
Summary: An adaptive mesh morphogenesis method is proposed for coarsening arbitrary unstructured meshes, utilizing a posteriori error estimation and an edge straightening scheme. The method can be recursively conducted, regardless of element type and mesh generation counting. Employing a topology-based data structure to handle mesh modification events, it effectively handles mesh coarsening for arbitrarily shaped elements while capturing problematic regions with sharp gradients or singularity.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Mathematics, Applied
Stefano Berrone, Alessandro D'Auria
Summary: This paper discusses the use of polygonal mesh refinement to solve two common issues and proposes a new refinement method for convex cells.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2022)
Article
Mathematics, Applied
Stefano Berrone, Andrea Borio, Alessandro D'Auria
Summary: The paper discusses discretization methods based on polygonal and polyhedral elements for differential problems on complex geometrical domains, and investigates the impact of polygonal refinement strategies on mesh quality and convergence rate. Experimental results on a geometrically complex geophysical problem validate the effectiveness of the proposed polygonal refinement strategies.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2021)
Article
Mathematics, Applied
Yang Liu, Shi Shu, Huayi Wei, Ying Yang
Summary: The article proposes a virtual element method to numerically approximate the Poisson-Nernst-Planck equations, and shows that the method performs well on general polygonal elements.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Ying Wang, Gang Wang, Yue Shen
Summary: We propose a new pressure-robust virtual element method for the incompressible Navier-Stokes equation and analyze its properties. This method introduces a reconstruction operator that maps the virtual function space to the piecewise RT finite element space, with extra conditions enforced. The well-posedness and convergence analyses are carried out, showing that the velocity error is not affected by the continuous pressure. Numerical examples are provided to support the theoretical findings, and the implementation of the reconstruction operator is demonstrated in detail.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Shipeng Xu
Summary: This paper derives a posteriori error estimates for the Weak Galerkin finite element methods for second order elliptic problems in terms of an H1-equivalent energy norm. The error analysis of the methods is proven to be valid for polygonal meshes under general assumptions, making it possible to solve Stokes equations and biharmonic equations on such meshes. The theoretical findings are verified by numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Sergio Salinas-Fernandez, Nancy Hitschfeld-Kahler, Alejandro Ortiz-Bernardin, Hang Si
Summary: This paper presents an algorithm for generating a new type of polygonal mesh from triangulations. The algorithm, called Polylla, divides the process into three phases to label edges, build polygons from terminal-edge regions, and transform non-simple polygons into simple ones. The resulting mesh contains both convex and non-convex shapes. Compared to the commonly used Voronoi-based meshes, Polylla meshes have fewer polygons and a simpler and faster generation algorithm. The validity of Polylla meshes is confirmed through experiments and numerical performance comparisons with Voronoi meshes using the virtual element method.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
M. Arrutselvi, D. Adak, E. Natarajan, S. Roy, S. Natarajan
Summary: This article proposes a method for discretizing nonlocal coupled parabolic problems within the framework of the virtual element method. By introducing an equivalent formulation, the computation of the Jacobian matrix is simplified and the sparsity is improved. The theoretical results are validated through numerical experiments.
Article
Mathematics, Applied
Ying Wang, Gang Wang, Feng Wang
Summary: This paper presents and analyzes a residual-type a posteriori error estimator for low-order virtual element discretization for the Stokes and Navier-Stokes problems, proving its globally upper and locally lower bounds for discretization error, with modifications for small viscosity cases. The effectiveness and flexibility of the designed error estimator combined with adaptive mesh refinement are verified through a series of benchmark tests.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Meihua Sheng, Di Yang, Zhiming Gao
Summary: In this paper, a conservative scheme based on the virtual element method is proposed to solve convection-diffusion problems on general meshes. The new scheme has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Mahdi Jabbari, Hamid Moslemi
Summary: The adaptive finite element method aims to achieve the desired accuracy with fewer degrees of freedom by refining the mesh to distribute the error uniformly. Polygonal elements have gained attention due to their flexibility in complex geometries, with the Voronoi diagram being the most common approach for creating a polygonal mesh. However, the inconsistency between the generated mesh and the desired mesh density is a drawback. In this study, an iterative algorithm is used to optimize a target error function based on the generated and desired mesh densities at each Gauss point, while clustering and additional modification techniques are employed to improve the mesh quality.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2022)
Article
Mathematics, Applied
Cesar Herrera, Ricardo Corrales-Barquero, Jorge Arroyo-Esquivel, Juan G. Calvo
Summary: This paper presents a numerical implementation of the Virtual Element Method with high order spaces. The convergence of the method is verified for different polygonal partitions. Additionally, a mesh-free application based on high-order polynomial space projection to approximate harmonic functions is discussed, which significantly reduces the computational cost.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
M. Arrutselvi, E. Natarajan
Summary: In this paper, the virtual element formulation for the nonlinear convection-diffusion-reaction equation is discussed, along with the use of the Streamline upwind Petrov-Galerkin stabilization method to reduce non-physical oscillations in the solution. The nonlinear term is evaluated after modification with a polynomial projection operator. Error analysis is performed to show the existence of the solution and derive convergence estimates in the energy norm, with numerical experiments conducted to validate the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Yanhui Zhou, Yanlong Zhang, Jiming Wu
Summary: In this paper, a polygonal finite volume element method (PFVEM) is proposed and analyzed for solving the anisotropic diffusion equation on convex polygonal meshes, based on the Wachspress generalized barycentric coordinates. The PFVEM reduces to the classical P-1-FVEM on triangular meshes but is not identical to the classical Q(1)-FVEM on quadrilateral meshes. The paper provides a new proof for Proposition 8 in [19], a crucial result for the derivation of interpolation error estimates. Furthermore, the H-2 error estimate of the Wachspress interpolation is proven for the error analysis of the PFVEM, and the optimal H-1 error estimate for the finite volume element solution is obtained under the coercivity assumption. Several numerical examples are presented to demonstrate the efficiency and robustness of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Wen-ming He, Hailong Guo
Summary: This study investigates the maximum norm error estimations for virtual element methods, establishing higher local regularity and optimal convergence results through analysis of Green's functions and high-order local error estimations for the virtual element solutions. The theoretical discoveries are validated with a numerical example on general polygonal meshes.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Dibyendu Adak, E. Natarajan, Sarvesh Kumar
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2019)
Article
Mathematics, Applied
M. Arrutselvi, D. Adak, E. Natarajan, S. Roy, S. Natarajan
Summary: This article proposes a method for discretizing nonlocal coupled parabolic problems within the framework of the virtual element method. By introducing an equivalent formulation, the computation of the Jacobian matrix is simplified and the sparsity is improved. The theoretical results are validated through numerical experiments.
Article
Engineering, Multidisciplinary
Dibyendu Adak, David Mora, Ivan Velasquez
Summary: In this work, the C0-nonconforming VEM is studied for fourth-order eigenvalue problems of thin plates with clamped boundary conditions on general shaped polygonal domain. The convergence analysis in discrete H2 seminorm and H1, L2 norms for both problems is derived by employing the enriching operator. The introduced schemes are shown to provide well approximation of the spectrum using the Babuska-Osborn spectral theory, and optimal order of rate of convergence for eigenfunctions and double order of rate of convergence for eigenvalues are proved. Numerical results are presented to demonstrate the good performance of the method on different polygonal meshes.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Dibyendu Adak
Summary: In this article, the mixed virtual element formulation is proposed to solve the nonlocal parabolic problem. A priori error estimates for both semi-discrete and fully-discrete schemes are derived and analyzed. The existence and uniqueness of the fully-discrete scheme are proven using Brouwer's fixed point argument. A series of numerical examples are provided to validate the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Dibyendu Adak, David Mora, Sundararajan Natarajan, Alberth Silgado
Summary: The study introduces a new Virtual Element Method of arbitrary order for the time dependent Navier-Stokes equations, which shows optimal convergence in both space and time variables. The theoretical analysis is validated through numerical examinations on four benchmark examples.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2021)
Article
Engineering, Multidisciplinary
Dibyendu Adak, E. Natarajan, Sarvesh Kumar
INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING SCIENCES AND APPLIED MATHEMATICS
(2016)
