Article
Mathematics, Applied
Sander Rhebergen, Garth N. Wells
Summary: This paper introduces a new preconditioner for the recently developed HDG method for finite element discretization of the Stokes equations. The new preconditioner reduces the number of globally coupled degrees-of-freedom and improves convergence speed, conservation properties, and CPU time compared to previous preconditioners. Numerical examples in two and three dimensions demonstrate the effectiveness of the method.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Mustafa E. Danis, Jue Yan
Summary: This study proposes a new formula for the nonlinear viscous numerical flux and extends it to the compressible Navier-Stokes equations using the direct discontinuous Galerkin method with interface correction (DDGIC). The new method simplifies the implementation and enables accurate calculation of physical quantities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Lin Mu, Xiu Ye, Shangyou Zhang
Summary: Pressure-robustness is crucial for incompressible fluid simulations. Enhancements to the discontinuous Galerkin finite element methods in the primary velocity-pressure formulation for solving Stokes equations have been developed to achieve pressure-robustness. The new schemes show improvements in source term modifications and have been validated through numerical experiments. Optimal-order error estimates have been established for the numerical approximations in various norms.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Aaron Baier-Reinio, Sander Rhebergen, Garth N. Wells
Summary: This paper presents an analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem under minimal regularity assumptions on the exact solution. The methods are shown to produce H(div)-conforming and divergence-free approximate velocities. A priori error estimates for the velocity that are independent of the pressure are derived, assuming only H1+s-regularity of the exact velocity fields for any s is an element of [0, 1]. Error estimates for the velocity and pressure in the L-2-norm are also obtained in this minimal regularity setting. The theoretical findings are supported by numerical computations.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Multidisciplinary
Manuel A. Sanchez, Shukai Du, Bernardo Cockburn, Ngoc-Cuong Nguyen, Jaime Peraire
Summary: In this paper, several high-order accurate finite element methods for the Maxwell's equations are presented, which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to the total energy. These methods are devised by taking advantage of the Hamiltonian structures of the Maxwell's equations and using spatial and temporal discretization techniques to ensure the conservation properties and convergence of the methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Dongwook Shin, Youngmok Jeon, Eun-Jae Park
Summary: This article introduces and analyzes arbitrary-order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier-Stokes equations. The study proves the injectivity of the lifting operator associated with trace variables for any polynomial degree, and obtains optimal error estimates in the energy norm by introducing nonstandard projection operators for the hybrid DG method. The numerical results presented in the article validate the theory and demonstrate the performance of the algorithm.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Engineering, Multidisciplinary
Manuel A. Sanchez, Bernardo Cockburn, Ngoc-Cuong Nguyen, Jaime Peraire
Summary: This paper presents a class of high-order finite element methods that conserve linear and angular momenta as well as energy for equations of linear elastodynamics by exploiting and preserving the Hamiltonian structure. Experimental results confirm optimal convergence and conservation properties of these methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Waldemar Rachowicz, Adam Zdunek, Witold Cecot
Summary: The DPG method is used for simulating three-dimensional compressible viscous flows, with stable schemes constructed for problems with small perturbation parameters. The approach uses weak formulation and specially designed optimal test functions, with a built-in a posteriori error estimation and mesh adaptivity for resolving reliable viscous fluxes in simulations of viscous flows.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Mathias Anselmann, Markus Bause
Summary: A new fictitious domain method based on higher order space-time finite element discretizations is proposed for simulating the nonstationary, incompressible Navier-Stokes equations on evolving domains. The method shows potential in efficiently handling intersections of moving domain boundaries with background mesh, weak formulation of Dirichlet boundary conditions, integration over cell intersections, and extension of physical quantities to the computational background mesh. The approach demonstrates optimal convergence properties in numerical experiments and considers well-known benchmarks such as flow around a cylinder and obstacles with cut cells.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Mathematics, Applied
Jing Wen, Jian Su, Yinnian He, Zhiheng Wang
Summary: This study presents a numerical finite element discretization method with strong mass conservation for the coupled Stokes and dual-porosity model. By utilizing different finite element spaces and discretization methods, the governing equations of Stokes region and dual-porosity domain are discretized to demonstrate the well-posedness of the discrete scheme and error estimates.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
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
Aikaterini Aretaki, Efthymios N. Karatzas, Georgios Katsouleas
Summary: This work presents an analysis of an unfitted discontinuous Galerkin discretization method for solving the Stokes system. The method utilizes high-order discontinuous velocities and pressures and combines advantages of accurate approximation and flexibility in handling complex geometries. The approach includes a fictitious domain framework, pressure stabilization, and ghost penalty strategies to enhance stability and robustness. The study investigates inf-sup stability, convergence order, and sensitivity to cut configuration, and provides numerical examples to verify the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Maojun Li, Yongping Cheng, Jie Shen, Xiangxiong Zhang
Summary: In this study, numerical schemes for the incompressible Navier-Stokes equations with variable density are investigated, focusing on preserving density bounds. It is found that a combination of continuous finite element method for momentum evolution and bound-preserving DG method for density evolution can achieve high accuracy and boundedness. Numerical tests demonstrate the effectiveness of the proposed scheme in various representative examples.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Tale Bakken Ulfsby, Andre Massing, Simon Sticko
Summary: We propose a novel cut discontinuous Galerkin (CutDG) method for solving stationary advection-reaction problems on surfaces embedded in Rd. The approach involves embedding the surface into a full-dimensional background mesh and using discontinuous piecewise polynomials as test and trial functions. By introducing a suitable stabilization technique, we are able to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. Numerical examples validate our theoretical findings.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Aycil Cesmelioglu, Jeonghun J. Lee, Sander Rhebergen
Summary: We introduce and analyze a hybridizable discontinuous Galerkin finite element method for the coupled Stokes-Biot problem. The method has the property that the discrete velocities and displacements satisfy the compressibility equations pointwise on the elements. We prove well-posedness of the discretization and provide a priori error estimates that demonstrate the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence for all unknowns and locking-free discretization.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
