Article
Mathematics, Applied
Limin Ma
Summary: In this paper, a unified analysis of superconvergence property for a large class of mixed discontinuous Galerkin methods is presented. This analysis is applicable to the Poisson equation and linear elasticity problems with symmetric stress formulations. Numerical experiments validate the effectiveness of locally postprocess schemes in improving displacement accuracy.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Guosheng Fu, Christoph Lehrenfeld, Alexander Linke, Timo Streckenbach
Summary: This study discusses the issues of volume-locking and gradient robustness in linear elasticity, proposing novel Hybrid Discontinuous Galerkin methods for discretization. By utilizing divergence-conforming discretization and additional optimization measures, effective control over gradient fields and spurious displacements is achieved.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Fleurianne Bertrand, Henrik Schneider
Summary: Existing a priori convergence results of the discontinuous Petrov-Galerkin method to solve the problem of linear elasticity are improved by using duality arguments, which lead to higher convergence rates for the displacement. Post-processing techniques are introduced to prove superconvergence, and numerical experiments are conducted to confirm the theory.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Gang Chen, Xiaoping Xie, Youcai Xu, Yangwen Zhang
Summary: This paper proposes and analyzes a stabilized embedded discontinuous Galerkin method for linear elasticity problems. A stabilized term is added to ensure locking-free approximations. The theoretical results are verified through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yanjun Li, Hai Bi
Summary: In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Boqian Shen, Beatrice Riviere
Summary: We propose a numerical method for solving the two-phase flow poroelasticity equations using the interior penalty discontinuous Galerkin method and a sequential time-stepping method. The existence of the solution is proved, and three-dimensional numerical results demonstrate the accuracy and robustness of the proposed method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Son-Young Yi, Sanghyun Lee, Ludmil Zikatanov
Summary: In this paper, we propose a locking-free enriched Galerkin method for solving the linear elasticity problem. The method not only achieves high accuracy and robustness, but also avoids volumetric locking when modeling incompressible materials.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
David A. Kopriva, Gregor J. Gassner
Summary: A hybrid continuous and discontinuous Galerkin spectral element approximation is proposed, utilizing continuous Galerkin on interior element faces and discontinuous Galerkin at physical boundaries or faces with property jumps. The method employs split form equations and two-point fluxes for stability on unstructured curved element meshes, while maintaining conservation and constant state preservation. Spectral accuracy is achieved for examples involving wave scattering at discontinuous medium boundaries.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Ruishu Wang, Zhuoran Wang, Jiangguo Liu
Summary: This paper presents a family of new weak Galerkin finite element methods for solving linear elasticity in the primal formulation. These methods use vector-valued polynomials of degree k >= 0 to approximate the displacement independently in element interiors and on edges of a convex quadrilateral mesh. The new methods do not require penalty or stabilizer and are free of Poisson-locking, while achieving optimal order (k + 1) convergence rates in displacement, stress, and dilation. Numerical experiments on popular test cases demonstrate the theoretical estimates and efficiency of these new solvers. The extension to cuboidal hexahedral meshes is briefly discussed.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Soeren Bartels, Andrea Bonito, Philipp Tscherner
Summary: An interior penalty discontinuous Galerkin method is proposed to approximate minimizers of a linear folding model using discontinuous isoparametric finite element functions with an approximation of a folding arc. The numerical analysis includes an a priori error estimate for an accurate folding curve representation. Additional estimates show that geometric consistency errors can be controlled separately with piecewise polynomial curve approximation of the folding arc. Numerical experiments validate the a priori error estimate for the folding model.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Hui Peng, Ruishu Wang, Xiuli Wang, Yongkui Zou
Summary: In this paper, the weak Galerkin finite element method is applied to a linear elasticity interface model. The model is discretized using double-valued weak functions on the interface, and a weak Galerkin method with single-valued functions on the interface is constructed for theoretical analysis and algorithm implementation. The well-posedness of the weak Galerkin scheme is proven, and a priori error estimates in energy norm and L2 norm are derived. Numerical experiments demonstrate the efficiency and locking-free property of the method.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Engineering, Multidisciplinary
Helene Barucq, Julien Diaz, Rose-Cloe Meyer, Ha Pham
Summary: The study utilizes the HDG method to numerically solve the two-dimensional anisotropic poroelastic wave equations, inheriting the advantages of the discontinuous Galerkin method without drastic increase in degrees of freedom. Through comparisons with analytical solutions and sensitivity analysis of stabilization parameters, the method's accuracy and practicality are confirmed.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Mathematics, Applied
Yue Wang, Fuzheng Gao, Jintao Cui
Summary: A new conforming discontinuous Galerkin method is studied for linear elasticity interface problems with discontinuous coefficients and displacement. This method, based on a new definition of weak gradient operator, does not require a stabilizer and employs a different weak divergence operator than the traditional weak Galerkin finite element method, resulting in reduced computational cost. Error estimates in discrete L-2 and H-1 norms are established with optimal order. Numerical examples validate the efficiency, accuracy, and locking-free property of the numerical method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Sanghyun Lee, Son-Young Yi
Summary: This paper presents a new coupled enriched Galerkin scheme for Biot's poroelasticity model based on the displacement-pressure formulation. The goal is to provide a stable and robust numerical method for a wide range of physical and numerical parameters. The method utilizes enriched linear Lagrange spaces as finite-dimensional solution spaces and achieves the inf-sup condition between the spaces by adding a stabilization term. The resulting coupled EG method is locally conservative and produces stable solutions without spurious oscillations or overshoots/undershoots. Well-posedness and optimal a priori error estimates are established, and numerical results in various scenarios are provided.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Linshuang He, Minfu Feng, Jun Guo
Summary: In this paper, we propose and analyze an H(div) conforming CDG method for the three-field Biot's consolidation model with displacement reconstruction technique. The method utilizes kth-order Brezzi-Douglas-Marini element for discretizing the displacement, and k-1th-order Raviart-Thomas-Nedelec element pairs for approximating the fluid flux and pore pressure. The H(div) CDG method is derived from the H(div) conforming formulation by replacing the gradient operator with a weak gradient operator, and optimal a-priori error estimates are proven for both semi-discrete scheme and fully-discrete scheme with backward Euler discretization in time.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
