Article
Mathematics, Applied
Jiming Yang, Yifan Su
Summary: An incompressible miscible displacement problem was investigated, and a two-grid algorithm for the problem was proposed. The analysis showed that the algorithm achieved asymptotically optimal approximation with less time spent, which was verified by numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou, Cunyun Nie
Summary: This paper discusses the application of discontinuous Galerkin approximations to the compressible miscible displacement problem. A two-grid algorithm is proposed, consisting of one coarse grid space and one fine grid space. Error estimates for concentration and velocity are presented, showing that the two-grid method achieves optimal approximations under certain conditions. Numerical results confirm the effectiveness of the algorithm.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou, Yifan Su
Summary: This paper studies an incompressible miscible displacement problem in porous media and proposes a two-grid algorithm based on an interior penalty discontinuous Galerkin method. With this algorithm, solving a nonlinear system on the discontinuous finite element space is reduced to solving a nonlinear problem on a coarse grid and a linear problem on a fine grid. The error estimate for the concentration in H-1-norm and the error estimate for the velocity in L-2-norm are obtained, and numerical experiments are provided to confirm the theoretical analysis.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou
Summary: This paper proposes a two-grid algorithm based on the Newton iteration method for modeling a compressible miscible displacement problem in porous media, using a combined mixed finite element and discontinuous Galerkin approximation. Error estimates in the H-1 norm for concentration and the L-2 norm for velocity are derived, showing that an asymptotically optimal approximation rate can be achieved with the two-grid algorithm if h = O(H-2) is satisfied. Numerical experiments demonstrate the effectiveness of the algorithm, consistent with the theoretical analysis.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Caixia Nan, Huailing Song
Summary: This paper presents a numerical method for two-phase miscible flow in porous media, utilizing the local discontinuous Galerkin method and IMEX-RK method. The method achieves second-order time discretization and optimal convergence analysis for pressure and concentration in L2-norm, demonstrated through numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jeonghun J. Lee, Omar Ghattas
Summary: In this paper, we propose a variant of enriched Galerkin methods for second order elliptic equations with over-penalization of interior jump terms and prove that optimal a priori error estimates with the standard discrete energy norm can be obtained by combining a priori and a posteriori error analysis techniques. We also show that the interior over-penalization is advantageous for constructing preconditioners robust to mesh refinement by analyzing the spectral equivalence of bilinear forms.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Waixiang Cao
Summary: In this paper, a new class of spectral volume (SV) methods are proposed, analyzed, and implemented for diffusion equations. The construction of control volumes based on four types of special points leads to four different SV schemes. A framework for the stability analysis and error estimates of these schemes is established, with discussions on the influence of numerical flux parameters on convergence rate and optimal coefficient choices. Numerical experiments demonstrate the stability and accuracy of the SV schemes for linear and nonlinear diffusion equations.
JOURNAL OF SCIENTIFIC COMPUTING
(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
Hui Bi, Mengyuan Zhang
Summary: This paper investigates the stability and error estimates of a fully discrete local discontinuous Galerkin method for solving the linear bi-harmonic equation. By introducing methods such as generalized alternating numerical fluxes and reference functions, the paper achieves unconditional stability and optimal error estimates for the fully discrete scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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
Haitao Leng
Summary: In this paper, the hybridized, embedded-hybridized and embedded discontinuous Galerkin methods for the Stokes equations with Dirac measures are analyzed. The velocity, velocity traces, and pressure traces are approximated by polynomials of degree k >= 1 and the pressure is discretized by polynomials of degree k - 1. The discrete velocity field satisfies a property called divergence-free, and the discrete velocity fields obtained by hybridized and embedded-hybridized discontinuous Galerkin methods are H(div)-conforming. A priori and a posteriori error estimates for the velocity in L-2-norm are obtained using duality argument and Oswald interpolation. Additionally, a posteriori error estimates for the velocity in W-1,W-q-seminorm and the pressure in L-q-norm are derived. Numerical examples are provided to validate the theoretical analysis and demonstrate the performance of the obtained a posteriori error estimators.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Soeren Bartels, Andrea Bonito, Philipp Tscherner
Summary: An interior penalty discontinuous Galerkin method is proposed to approximate minimizers of a linear folding model using discontinuous isoparametric finite element functions with an approximation of a folding arc. The numerical analysis includes an a priori error estimate for an accurate folding curve representation. Additional estimates show that geometric consistency errors can be controlled separately with piecewise polynomial curve approximation of the folding arc. Numerical experiments validate the a priori error estimate for the folding model.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Lulu Tian, Hui Guo, Rui Jia, Yang Yang
Summary: This paper investigates the application of local discontinuous Galerkin methods to compressible wormhole propagation with a Darcy-Forchheimer model. By addressing various theoretical challenges, stability and error estimates of the scheme are proven, followed by numerical experiments to verify the results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Soren Bartels
Summary: A class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems have been devised and analyzed. The derived discrete duality relations lead to optimal error estimates for problems involving total-variation regularization or obstacles. The analysis provides explicit estimates that accurately determine the role of stabilization parameters, and numerical experiments confirm the optimality of the estimates.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Florian Kummer, Jens Weber, Martin Smuda
Summary: The software package BoSSS discretizes partial differential equations with discontinuous coefficients and/or time-dependent domains using an eXtended Discontinuous Galerkin (XDG) method. This work introduces the XDG method, develops a formal notation capturing important numerical details, and presents iterative solvers for extended DG systems.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
