Article
Mathematics, Applied
Yao Cheng, Xuesong Wang
Summary: This paper considers a local discontinuous Galerkin (LDG) method for a one-dimensional singularly perturbed convection-diffusion problem with an exponential boundary layer. Based on the technique of discrete Green's function, the optimal pointwise convergence (up to a logarithmic factor) of the LDG method is established on three typical families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type, and Bakhvalov-type. Numerical experiments are also provided.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Yao Cheng, Li Yan, Xuesong Wang, Yanhua Liu
Summary: In this study, the local discontinuous Galerkin (LDG) method with a generalized alternating numerical flux is proposed for a one-dimensional singularly perturbed convection-diffusion problem. The double-optimal local maximum-norm error estimate is derived on the quasi-uniform meshes for the first time. Additionally, the discrete maximum principle and global L1-error estimate established in the literature are improved.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jose Luis Gracia, Eugene O'Riordan
Summary: A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is studied, where a particular complimentary error function is identified to match the discontinuity. Numerical approximation is used to compare the analytical function with the solution of the problem, with a coordinate transformation for aligning a layer-adapted mesh. The numerical analysis shows that the method is parameter-uniform, with numerical results demonstrating the established pointwise error bounds.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Yaxiang Li, Jiangxing Wang
Summary: In this paper, a high order hybridizable discontinuous Galerkin method (HDG) on two layer-adapted meshes is developed for the singularly perturbed convection-diffusion problems. The existence and uniqueness of the HDG solutions are verified. The implementation of two different anisotropic meshes allows for uniform super-convergence in both one-dimensional and two-dimensional cases.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Yao Cheng
Summary: In this study, the local discontinuous Galerkin method is applied to a singularly perturbed problem with two parameters. The use of layer-adapted mesh and various projections results in robust or almost robust convergence in different norms. Careful approximation error estimation on anisotropic meshes leads to different error estimates in various norms, with numerical experiments conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Yao Cheng, Martin Stynes
Summary: This paper investigates a singularly perturbed convection-diffusion problem on the unit square in R-2, which has exponential and characteristic boundary layers in its solution. The problem is numerically solved using the local discontinuous Galerkin (LDG) method on Shishkin meshes. The convergence rate of the LDG solution to the true solution is proven to be ((N-1 ln N)(k+1/2)) O in an associated energy norm, uniformly in the singular perturbation parameter. Additionally, it is shown that the convergence order increases to O when measuring the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution.
NUMERISCHE MATHEMATIK
(2023)
Article
Mathematics, Applied
Jiaqi Li, Leszek Demkowicz
Summary: Building upon the standard Discontinuous Petrov-Galerkin (DPG) method in Hilbert spaces, this study generalizes the approach to Banach spaces. Numerical experiments on a 1D convection-dominated diffusion problem demonstrate that the Banach-based method yields solutions less affected by the Gibbs phenomenon. H-adaptivity is implemented using an error representation function as an indicator of error. Published by Elsevier Ltd.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Y. A. O. Cheng, S. H. A. N. Jiang, M. A. R. T. I. N. Stynes
Summary: This paper numerically solves a convection-diffusion problem on the unit square in Double-struck capital R2 with exponential boundary layers using the local discontinuous Galerkin (LDG) method. They establish the superconvergence property of the LDG solution on three types of layer-adapted meshes, which leads to an optimal bound for the L2 error. Numerical experiments confirm their theoretical results.
MATHEMATICS OF COMPUTATION
(2023)
Article
Mathematics, Applied
Xiaowei Liu, Min Yang, Jin Zhang
Summary: In this paper, the weak Galerkin method on a Shishkin rectangular mesh is analyzed for a singularly perturbed convection-diffusion problem in two dimensions. The method achieves supercloseness through a specially constructed interpolant, which consists of vertices-edges-element interpolant inside the layers and modified Gauss-Radau interpolant outside the layers in the interior of each element, and vertices-edges-element interpolant inside the layers and weighted L2 projection outside the layers on the boundary of each element. Additionally, over-penalization technique is used inside the layers, and supercloseness of order k + 1/2 is proved, even up to almost k + 1 under appropriate assumptions. Numerical experiments validate the theoretical result.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
J. L. Gracia, E. O'Riordan
Summary: In this study, a singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified to match the discontinuity in the initial condition and satisfy the homogenous parabolic differential equation associated with the problem. By using an upwind finite difference operator with a layer-adapted mesh, the numerical approximation of the difference between the analytical function and the solution of the parabolic problem is shown to be parameter-uniform, with numerical results illustrating the theoretical error bounds established in the paper.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Vivek Kumar, Guenter Leugering
Summary: This article focuses on the study of singular perturbed static convection-diffusion equations with varying coefficients on a metric graph G = (V, E). The emphasis is on the convection dominated situation where a small parameter epsilon > 0 appears in front of the diffusion term. The reduced problem in the limit epsilon -> 0 may exhibit boundary layers at multiple vertices and simple nodes. Various scenarios are analyzed and validated in several test cases, using exemplary graphs and an upwind finite difference method on a piece-wise Shishkin mesh. Error estimates are discussed to demonstrate epsilon-uniform convergence. (c) 2023 Elsevier B.V. All rights reserved.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Gautam Singh, Srinivasan Natesan
Summary: The parabolic convection-diffusion-reaction problem is discretized using the NIPG method in space and the DG method in time. Piecewise Lagrange interpolation at Gauss points is used to improve the order of convergence, and the error bound in the discrete energy norm is estimated. The study demonstrates superconvergence properties of the DG method with (k+1)-order convergence in space and (l+1)-order convergence in time, with numerical results confirming the theoretical findings.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
J. L. Gracia, E. O'Riordan
Summary: This article investigates a singular perturbed parabolic problem of convection-diffusion type with incompatible inflow boundary and initial conditions. When the coefficients are constant, a set of singular functions is identified to match the incompatibilities in the data and satisfy the associated homogeneous differential equation. In the case of variable coefficients and continuous boundary/initial data, a numerical method is developed with its convergence rate depending on the level of compatibility satisfied by the data. Numerical results are provided to validate the theoretical error bounds for both approaches.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Natalia Kopteva, Richard Rankin
Summary: The symmetric interior penalty discontinuous Galerkin method and its weighted averages version are applicable on shape-regular nonconforming meshes for solving singularly perturbed semilinear reaction-diffusion equations. Residual-type a posteriori error estimates in maximum norm are given, with error cons...
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer, Ernst P. Stephan
IMA JOURNAL OF NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer, Antti H. Niemi
MATHEMATICS OF COMPUTATION
(2019)
Article
Engineering, Multidisciplinary
Thomas Fuhrer, Norbert Heuer
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2019)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Editorial Material
Mathematics, Applied
Fleurianne Bertrand, Leszek Demkowicz, Jay Gopalakrishnan, Norbert Heuer
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Michael Karkulik, Jens Markus Melenk
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2019)
Article
Mathematics, Applied
Thomas Fuhrer, Alexander Haberl, Norbert Heuer
Summary: The paper explores trace operators and spaces related to the bi-Laplacian, aiming to establish well-posed formulations under low regularity conditions assuming an L-2 right-hand side function. By defining traces and integration-by-parts formulas, two well-posed formulations are obtained using different combinations of trace operators, with numerical experiments conducted to validate the results. The analysis applies to space dimensions larger than or equal to two, with exceptions noted in an appendix regarding the nonclosedness of a trace space.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer, Francisco-Javier Sayas
NUMERISCHE MATHEMATIK
(2020)
Article
Mathematics, Applied
Thomas Fuhrer, Carlos Garcia Vera, Norbert Heuer
Summary: The study introduces a numerical solution method for beam bending models based on DPG method, capable of accurately approximating transverse deflection and bending moment under various boundary conditions. Numerical results demonstrate its good convergence performance.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Michael Karkulik
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Thomas Fuhrer, Michael Karkulik
Summary: The method is based on residual minimization in space-time L-2 norms, ensuring that the resulting bilinear form is symmetric and coercive, leading to uniformly stable conforming discretization. Stiffness matrices are symmetric, positive definite, and sparse, with a local a-posteriori error estimator available for free. The approach also features full space-time adaptivity, with a priori error analysis conducted on highly structured simplicial space-time meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Leszek Demkowicz, Thomas Fuhrer, Norbert Heuer, Xiaochuan Tian
Summary: This paper efficiently implements the double adaptivity algorithm of Cohen et al. (2012) in the Petrov-Galerkin method with optimal test functions. The method is demonstrated to be feasible in the context of convection-dominated diffusion problems and is applicable to various well-posed first-order partial differential equation systems.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer
Summary: This study investigates the lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincare-type inequality, and demonstrates how robustness can be re-established by using appropriately scaled test norms. By studying the Poisson problem and the Kirchhoff-Love plate bending model, numerical experiments confirm the findings, including cases with an-isotropic domains and mixed boundary conditions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer, Michael Karkulik
Summary: In this study, optimal error estimates are analyzed for a primal DPG formulation of a class of parabolic problems using backward Euler time stepping schemes. For the heat equation, the solution of the primal DPG formulation is shown to equal the solution of a standard Galerkin scheme, resulting in optimal error bounds found in the literature. However, in the presence of advection and reaction terms, optimal error bounds analysis requires the use of elliptic projection operators focused on the spatial part of the PDE rather than the full PDE at a time step.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Thomas Fuhrer, Norbert Heuer
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2018)
