Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Stephen Metcalfe, Siva Nadarajah
Summary: In this work, a new quasi-optimal test norm for discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation is proposed. The theoretical analysis shows that the proposed test norm leads to favorable scalings between the target norm and the energy norm. Numerical experiments are conducted to confirm the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
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
Constantin Bacuta, Leszek Demkowicz, Jaime Mora, Christos Xenophontos
Summary: This work focuses on two problems: analyzing the DPG method in fractional energy spaces, and investigating a non-conforming version of the DPG method for general polyhedral meshes. The ultraweak variational formulation is used for the model Laplace equation, and theoretical estimates are supported by 3D numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Antti H. Niemi
Summary: We investigate the approximation properties of the primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. Our study extends the previous convergence results and duality arguments to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The theoretical findings are validated by numerical experiments, which also compare the primal DPG method with a conventional least squares finite element method with the same number of degrees of freedom.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Ankit Chakraborty, Georg May
Summary: We propose an anisotropic hp-mesh adaptation strategy for discontinuous Petrov-Galerkin (DPG) finite element schemes using a continuous mesh model with optimal test functions, extending previous work on h-adaptation. The strategy utilizes the residual-based error estimator of the DPG discretization to compute the polynomial distribution and anisotropy of the mesh elements. Local problems on element patches are solved to predict the optimal order of approximation, making these computations highly parallelizable. The performance of the strategy is demonstrated through numerical examples on triangular grids.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Leszek Feliks Demkowicz, Nathan Roberts, Judit Munoz-Matute
Summary: This paper studies both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. It determines that the common approach of constructing a local Fortin operator for DPG stability analysis is not feasible for this problem. Instead, a new approach based on direct proof of discrete stability is developed, along with the introduction of a subgrid mesh. The argument is supported by mathematical analysis and numerical experiments.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Simon Becher, Gunar Matthies
Summary: This paper presents a unified analysis for a family of variational time discretization methods applied to non-stiff initial value problems. The analysis includes discontinuous Galerkin methods and continuous Galerkin-Petrov methods, with a focus on global error and superconvergence properties under weak abstract assumptions. Numerical experiments support the theoretical results.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Zhaonan Dong, Emmanuil H. Georgoulis
Summary: A new variant of the IPDG method, called robust IPDG (RIPDG), is proposed, which involves weighted averages of the gradient of the approximate solution to enhance its robustness. Numerical experiments show that the RIPDG method performs better than the standard IPDG method in terms of error behavior and conditioning in scenarios with strong local variation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Eirik Valseth, Albert Romkes, Austin R. Kaul
Summary: The study proposes the numerical analysis of the Cahn-Hilliard equation using the AVS-FE method, which provides numerical stability and symmetry. By employing globally continuous spaces and optimal test functions, the AVS-FE method can solve the equation in both space and time without restrictive conditions. Results show optimal convergence rates and mesh adaptive refinements using an error estimator of the AVS-FE method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Hailiang Liu, Zhongming Wang, Peimeng Yin, Hui Yu
Summary: In this paper, we propose and analyze third order positivity-preserving discontinuous Galerkin schemes for solving the time-dependent system of Poisson-Nernst-Planck equations. Our method ensures the positivity of numerical solutions and restores it through a scaling limiter.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Tom Lewis, Aaron Rapp, Yi Zhang
Summary: This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson's problem by examining the relationship between the Laplacian and the discrete Laplacian. The DWDG methods are derived from the DG differential calculus framework, which replaces continuous differential operators with discrete ones. We establish error estimates and explore the relationship between the DWDG approximation and the Ritz projection. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
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
Computer Science, Interdisciplinary Applications
Brody R. Bassett, J. Michael Owen
Summary: The time-dependent, gray, linear radiation transport equation is discretized using the meshless local Petrov-Galerkin method with reproducing kernels, integrated using a Voronoi tessellation that automatically adjusts resolution and includes streamline-upwind Petrov-Galerkin stabilization. The angular quadrature can be selectively refined for higher angular resolution, with results indicating first-order convergence in time and second-order convergence in space using Krylov iterative methods for solving transport and scattering source.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Fleurianne Bertrand, Henrik Schneider
Summary: Existing a priori convergence results of the discontinuous Petrov-Galerkin method to solve the problem of linear elasticity are improved by using duality arguments, which lead to higher convergence rates for the displacement. Post-processing techniques are introduced to prove superconvergence, and numerical experiments are conducted to confirm the theory.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Dietmar Gallistl
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Applied
Daniele Boffi, Dietmar Gallistl, Francesca Gardini, Lucia Gastaldi
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Interdisciplinary Applications
D. Gallistl, D. Peterseim
MULTISCALE MODELING & SIMULATION
(2017)
Article
Mathematics, Applied
D. Gallistl, P. Huber, D. Peterseim
NUMERISCHE MATHEMATIK
(2017)
Article
Mathematics, Applied
Dietmar Gallistl
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2017)
Article
Mathematics, Applied
Dietmar Gallistl
MATHEMATICS OF COMPUTATION
(2019)
Article
Mathematics, Applied
Dietmar Gallistl
MATHEMATICS OF COMPUTATION
(2019)
Article
Mathematics, Applied
Dietmar Gallistl, Patrick Henning, Barbara Verfuerth
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Carsten Carstensen, Dietmar Gallistl, Yunqing Huang
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2018)
Article
Mathematics, Applied
Carsten Carstensen, Dietmar Gallistl, Joscha Gedicke
NUMERISCHE MATHEMATIK
(2019)
Article
Mathematics, Applied
Dietmar Gallistl, Endre Suli
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Dietmar Gallistl, Mira Schedensack
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2020)
Article
Mathematics, Interdisciplinary Applications
Dietmar Gallistl, Timo Sprekeler, Endre Suli
Summary: The paper introduces a mixed finite element method for approximating the periodic strong solution to the fully nonlinear second-order Hamilton-Jacobi-Bellman equation with coefficients satisfying the Cordes condition. The second part of the paper focuses on numerically homogenizing such equations and approximating the effective Hamiltonian. Numerical experiments demonstrate the effectiveness of the approximation scheme for the effective Hamiltonian and the numerical solution for the homogenized problem.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Mathematics, Applied
Dietmar Gallistl, Daniel Peterseim
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2019)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Donald L. Brown, Dietmar Gallistl, Daniel Peterseim
MESHFREE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS VIII
(2017)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
