Article
Mathematics, Applied
Jiming Yang, Jing Zhou, Hongbin Chen
Summary: A full-discrete two-grid discontinuous Galerkin approximation is proposed for nonlinear parabolic problems. The L-2-norm error analysis of the two-grid method is carried out, showing that the algorithm achieves an asymptotically optimal approximation when mesh sizes satisfy h = O(H-2). Numerical examples are presented to validate the efficiency of the algorithm.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Thomas Bohlen, Mario Ruben Fernandez, Johannes Ernesti, Christian Rheinbay, Andreas Rieder, Christian Wieners
Summary: This paper presents a new framework for full waveform inversion in the visco-acoustic regime, using a novel combination of the Discontinuous Galerkin method and the inverse solver. Successful reconstructions in a simplified cross-well setting demonstrate the applicability of this new approach, which has not been previously applied in the context of FWI.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Chanjie Li, Kai Qu
Summary: We investigate the reformulated two-dimensional (2-D) perfectly matched layer (PML) models based on the original 3-D PML model developed by Cohen and Monk in 1999. We propose the discontinuous Galerkin methods for solving both 2-D TMz and TEz models, and establish the proofs of stability and error estimate. The numerical results demonstrate the accuracy and performance of our method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Hector Navarro-Garcia, Ruben Sevilla, Enrique Nadal, Juan Jose Rodenas
Summary: The Cartesian grid discontinuous Galerkin finite element method combines the accuracy and efficiency of high-order discontinuous Galerkin discretization with the simplicity of a Cartesian mesh. Special treatment is required for elements intersecting the physical domain boundary to minimize their impact on the algorithm's performance. By implementing a stabilization procedure, unstable degrees of freedom are eliminated and supporting regions of their shape functions are reassigned to neighboring elements, improving the method's stability and accuracy.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Mathematics, Applied
Norikazu Saito
Summary: The study focuses on the discontinuous Galerkin (DG) time-stepping method for the abstract evolution equation of parabolic type, establishing the inf-sup condition for the DG bilinear form. Through appropriate regularity assumptions and direct applications of standard interpolation error estimates, optimal order error estimates are obtained successfully.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Liuqiang Zhong, Liangliang Zhou, Chunmei Liu, Jie Peng
Summary: This paper studies the two-grid interior penalty discontinuous Galerkin (IPDG) method for mildly nonlinear second-order elliptic partial differential equations. The well-posedness of the IPDG finite element discretizations is established by introducing the equivalent weak formulation and combining Brouwer's fixed point theorem. Error estimates for the discrete solution in various norms are derived, and a two-grid method is designed for solving the IPDG discretization scheme with corresponding error estimates provided. Numerical experiments confirm the efficiency of the proposed approach.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
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
Paola F. Antonietti, Francesca Bonizzoni, Marco Verani
Summary: This paper presents a novel space-time discretization method for the linear scalar-valued dissipative wave equation. The structured approach combines the advantages of both the Virtual Element (VE) discretization in space and the Discontinuous Galerkin (DG) method in time. The proposed scheme is implicit and has been proven to be unconditionally stable and accurate in space and time.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Shashank Jaiswal
Summary: Adaptivity is crucial for addressing practical challenges, especially in computational fluid dynamics workflow. The mixed non-conforming discontinuous Galerkin discretization method is introduced for the full Boltzmann equation, providing optimal convergence for non-linear kinetic systems on non-orthogonal grids. The method allows for analysis of complex problems on massively parallel scales and is applicable to a wide range of rarefied flows. The computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible compared to conforming domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Xin Liu
Summary: In this paper, a local discontinuous Galerkin (LDG) method is proposed to simulate wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proven for both semi-discrete and full-discrete LDG schemes. Numerical results justify the theoretical analysis and demonstrate the interesting wave concentration property of the electromagnetic concentrator.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Mustafa Engin Danis, Jue Yan
Summary: In this study, a unified and general framework for direct discontinuous Galerkin methods is proposed. The framework eliminates the need for the antiderivative of the nonlinear diffusion matrix, allowing for a simple definition of the numerical flux. Nonlinear stability analyses and numerical experiments validate the performance of the new methods, showing that the symmetric and interface correction versions achieve optimal convergence and outperform the nonsymmetric version. Furthermore, the new direct discontinuous Galerkin methods accurately capture singular or blow up solutions.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Mathematics, Applied
Haijin Wang, Anping Xu, Qi Tao
Summary: This paper presents the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations, and analyzes the stability and error estimates of the corresponding fully discrete schemes by coupling with a specific Runge-Kutta type implicit-explicit time discretization. Numerical experiments are conducted to verify the theoretical results.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
O. Koshkarov, G. Manzini, G. L. Delzanno, C. Pagliantini, V. Roytershteyn
Summary: The paper discusses a spectral method for numerical simulation of the three-dimensional Vlasov-Maxwell equations, based on a spectral expansion of the velocity space using asymmetrically weighted Hermite functions. The resulting system of time-dependent nonlinear equations is discretized using the discontinuous Galerkin method in space and the method of lines for time integration. The developed code, SPS-DG, is successfully applied to standard plasma physics benchmarks, demonstrating accuracy, robustness, and parallel scalability.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Mathematics, Applied
Alex Kaltenbach, Michael Ruzicka
Summary: In this paper, a local discontinuous Galerkin approximation is proposed for fully nonhomogeneous systems of p-Navier--Stokes type. By using the primal formulation, the well-posedness, stability (a priori estimates), and weak convergence of the method are proved. A new discontinuous Galerkin discretization of the convective term is proposed, and an abstract nonconforming theory of pseudomonotonicity, which is applied to the problem, is developed. The approach is also used to treat the p-Stokes problem.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
