Article
Mathematics, Applied
Gregory Etangsale, Marwan Fahs, Vincent Fontaine, Nalitiana Rajaonison
Summary: In this paper, we improve the a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The theoretical results are supported by numerical evidence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Ming Tang, Xiaoqing Xing, Liuqiang Zhong
Summary: In this paper, a residual type a posteriori error estimator is designed for the mixed interior penalty discontinuous Galerkin method for H(curl)-elliptic problems. It is proven that the indicator is both reliable and efficient. Numerical experiments are conducted to validate the performance of the indicator within an adaptive mesh refinement procedure.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
D. Leykekhman
Summary: The paper aims to derive error estimates for biharmonic problems using the C-0 interior penalty method by employing dyadic decompositions of convex polygon domains. The proofs involve local energy estimates and new pointwise Green's function estimates for the continuous problem, which have independent significance.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Ying Liu, Yufeng Nie
Summary: This paper derives the a priori and a posteriori error estimates for the weak Galerkin finite element method with CrankNicolson time discretization applied to parabolic equations. The a priori estimates are based on existing results for elliptic projection problems, while the a posteriori estimates use an elliptic reconstruction technique to decompose the true error into elliptic and parabolic components. These estimates are further used to develop a temporal and spatial adaptive algorithm, with numerical results provided to validate the proposed estimators on uniform and adaptive meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics
Ying Sheng, Tie Zhang
Summary: This paper proposes a finite volume element method of primal-dual type to solve the ill-posed elliptic problem with lacking or overlapping boundary value condition. The well-posedness of the discrete scheme is proven and error estimations of the finite volume solution are derived. Numerical experiments are provided to verify the effectiveness of the proposed method.
Article
Mathematics, Applied
Gautam Singh, Srinivasan Natesan, Ali Sendur
Summary: This article focuses on the construction and analysis of a non-symmetric interior penalty Galerkin method (NIPG) on Shishkin mesh for solving singularly perturbed 2D elliptic boundary-value problems (BVPs). The piecewise Lagrange interpolation at Gaussian points is employed to improve the order of convergence of the interpolation error. The superconvergence properties of the NIPG method are studied, and O(N-1 ln N)(k+1) order of convergence is proven in the discrete energy-norm. Various numerical experiments are conducted to validate the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Applied
Mahboub Baccouch
Summary: This paper derives two a posteriori error estimates for the LDG method applied to linear second-order elliptic problems on Cartesian grids, showing the superconvergence of the LDG solution gradient and introducing a postprocessing gradient recovery scheme. The proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the L-2-norm under mesh refinement with p + 1 order of convergence. Additionally, a local adaptive mesh refinement procedure is presented based on local and global a posteriori error estimates, validated through numerical examples.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Will Thacher, Hans Johansen, Daniel Martin
Summary: We propose a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. The method demonstrates second, fourth, and sixth order accuracy on various tests, including problems with high contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We develop a generalized truncation error analysis and a simple method based on Green's theorem for computing exact geometric moments, which enable easy inclusion of spatially-varying coefficients and jump conditions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Yuxin Shang, Hongying Huang
Summary: This paper studies the hp-type error estimate of the direct discontinuous Galerkin (DDG) method for the second-order elliptic problem. It derives a priori error estimates with respect to the mesh size h and the polynomial degree p in energy norm and L-2 norm. The theoretical and numerical results show that the error convergent rates with respect to the mesh size h in the energy norm and L-2 norm are optimal, and the error with respect to the polynomial degree p in energy norm has a suboptimal convergence rate of half an order of p. Numerical examples demonstrate that the convergence rate about p is much greater than that obtained theoretically when the singular point lies in the interior of the element.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Guofang Chen, Junliang Lv, Xinye Zhang
Summary: This paper solves a second-order nonlinear elliptic equation using the finite volume element method and provides rigorous error estimates. The computational domain is divided into convex quadrilateral meshes. The isoparametric bilinear element space is chosen as the trial function space and the piecewise constant function space is chosen as the test function space. The boundedness and coercivity of the bilinear form are proved on the h(2)-parallelogram mesh, and the existence and uniqueness of the numerical solution are established using the Brouwer fixed point theorem. The paper also derives estimates for parallel to(del(u-u(h))parallel to and parallel to u-u(h)parallel to(0) under certain regularity assumptions. Numerical experiments on quadrilateral meshes are conducted to calculate the convergence orders in H-1 and L-2 norms, which align with the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Andreas Dedner, Jan Giesselmann, Tristan Pryer, Jennifer K. Ryan
Summary: This work examines a posteriori error control for post-processed approximations to elliptic boundary value problems, introducing a new class of post-processing operator to optimize various reconstruction operators. Extensive numerical tests are conducted to validate the analytical findings of the study.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
S. Nakov, I Toulopoulos
Summary: This paper focuses on finite element discretizations for quasilinear elliptic problems in divergence form, demonstrating existence and uniqueness of continuous and discrete problems. By deriving discretization error estimates under general regularity assumptions and using high order polynomial spaces, convergence rates are verified numerically. The key idea lies in carefully considering the relation between the natural W-1, W-p seminorm and a specific quasinorm, allowing for interpolation estimates in the quasinorm from known interpolation estimates in the W-1, W-p seminorm. Additionally, a simplified proof of known near-best approximation results in W-1, W-p seminorm is provided based on the corresponding result in the quasinorm mentioned.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Jiali Qiu, Fei Wang, Min Ling, Jikun Zhao
Summary: In this paper, an interior penalty virtual element method is developed to solve a Kirchhoff plate contact problem, and the well-posedness of the discrete problem is proved. Additionally, it is shown that the lowest-order VEM achieves optimal convergence order.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Zhaonan Dong, Emmanuil H. Georgoulis
Summary: A new variant of the IPDG method, called robust IPDG (RIPDG), is proposed, which involves weighted averages of the gradient of the approximate solution to enhance its robustness. Numerical experiments show that the RIPDG method performs better than the standard IPDG method in terms of error behavior and conditioning in scenarios with strong local variation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
R. H. W. Hoppe
Summary: We consider an adaptive C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of the fourth order von Karman equations with homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a six-field formulation of the finite element discretized von Karman equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 2,2 0 norm in terms of the associated primal and dual energy functionals. It requires the construction of equilibrated fluxes and equilibrated moment tensors which can be computed on local patches around interior nodal points of the triangulations. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.
APPLIED NUMERICAL MATHEMATICS
(2023)
