Article
Mathematics, Applied
Andreas Rupp, Moritz Hauck, Vadym Aizinger
Summary: The method introduced in this work generalizes the enriched Galerkin method with an adaptive two-mesh approach, proving stability and error estimates for a linear advection equation. The analysis technique allows for arbitrary degrees of enrichment on both coarse and fine meshes, covering a wide range of methods from continuous finite element to discontinuous Galerkin with local subcell enrichment. Numerical experiments confirm the analytical results and show good robustness of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
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
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
Engineering, Multidisciplinary
Kerstin Weinberg, Christian Wieners
Summary: We propose a new numerical approach for wave induced dynamic fracture. The method combines a discontinuous Galerkin approximation of elastic waves and a phase-field approximation of brittle fracture. The algorithm is staggered in time and uses an implicit midpoint rule for wave propagation and an implicit Euler step for phase-field evolution. Examples in two and three dimensions demonstrate the advantages of this approach in computing crack growth and spalling initiated by reflected and superposed waves.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
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
Mathematics
Petr Kubera, Jiri Felcman
Summary: This article deals with numerical solution of macroscopic models of pedestrian movement, focusing on the verification of the model. Two approaches for source term discretization and two splitting schemes for the numerical solution of the coupled system are proposed, leading to four different numerical methods for the Pedestrian Flow Equations. The comparative study of the numerical solutions shows that all proposed methods are in good agreement.
Article
Mathematics, Applied
Ivy Weber, Gunilla Kreiss, Murtazo Nazarov
Summary: This paper investigates the stability of a numerical method for solving the wave equation using matrix eigenvalue analysis to calculate time-step restrictions. It is found that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, while the restriction for symmetric interior penalty DG schemes is tighter. The best time-step restriction is obtained for continuous Hermite finite elements.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Materials Science, Multidisciplinary
Manas Vijay Upadhyay, Jeremy Bleyer
Summary: A time-explicit Runge-Kutta discontinuous Galerkin finite element scheme is proposed for solving the dislocation transport initial boundary value problem in 3D. This scheme provides stable and accurate numerical solutions through a combination of spatial and temporal discretization. Parameter study and simulation results show that the proposed scheme is more robust and accurate compared to existing methods based on continuous Galerkin finite element or fast Fourier transform.
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
(2022)
Article
Mathematics, Applied
Ahmed AL-Taweel, Lin Mu
Summary: A new upwind weak Galerkin finite element scheme for linear hyperbolic equations is developed in this paper, with an upwind-type stabilizer imposed. Error estimate in a suitable norm is investigated, and numerical examples are presented for validation of theoretical conclusions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Herbert Egger, Bogdan Radu
Summary: This study focuses on the numerical approximation of Maxwell's equations in the time domain using a second-order H(curl) conforming finite element approximation. A mass-lumping strategy and modification of the finite element space are proposed to achieve higher accuracy levels.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Jens Markus Melenk, Alexander Rieder
Summary: The study involves a time-dependent problem generated by a nonlocal operator in space. The approach includes spatial discretization using hp-finite elements and a Caffarelli-Silvestre extension, and time discretization using hp-discontinuous Galerkin based time stepping. Exponential convergence is proven in an abstract framework for the spatial domain Omega.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Federico Vismara, Tommaso Benacchio, Luca Bonaventura
Summary: This paper proposes and analyzes a seamless extended Discontinuous Galerkin (DG) discretization method for advection-diffusion equations on semi-infinite domains. The method splits the semi-infinite half line into a finite subdomain and a semi-unbounded subdomain, and uses different sets of basis functions in each subdomain. Numerical experiments show that the method has negligible errors and efficient performance. The extended framework is able to simulate absorbing boundary conditions without additional conditions at the interface.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Ruo Li, Qicheng Liu, Fanyi Yang
Summary: The proposed discontinuous least squares finite element method efficiently solves the indefinite time-harmonic Maxwell equations by minimizing the functional over piecewise polynomial spaces. This method is stable without any constraint on mesh size and demonstrates optimal convergence rate under the energy norm and sub-optimal convergence rate under the L-2 norm. Numerical results in two and three dimensions confirm the accuracy of error estimates.
IMA JOURNAL OF NUMERICAL ANALYSIS
(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
Computer Science, Interdisciplinary Applications
Bernard Kapidani, Lorenzo Codecasa, Joachim Schoeberl
Summary: The paper presents a new numerical method for solving the time-dependent Maxwell equations on unstructured meshes in two dimensions. The method incorporates arbitrary polynomial degrees and avoids the need for penalty parameters or numerical dissipation to achieve stability. It is shown to provide spurious-free solutions with expected convergence rates through numerical tests.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
