Article
Mathematics, Applied
Z. Dong, L. Mascotto
Summary: This paper proves the effectiveness of hp-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) in solving the biharmonic problem with homogeneous essential boundary conditions. The study considers both tensor product-type meshes in two and three dimensions, as well as triangular meshes in two dimensions. A key aspect of the analysis is the construction of a global H-2 piecewise polynomial approximants with hp-optimal approximation properties over the given meshes. The paper also discusses the hp-optimality of C-0-IPDG in two and three dimensions, as well as the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and show that p-suboptimality occurs in the presence of singular essential boundary conditions.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Christoph Erath, Lorenzo Mascotto, Jens M. Melenk, Ilaria Perugia, Alexander Rieder
Summary: In this paper, we present a coupling method that combines the discontinuous Galerkin finite element method with the boundary element method to solve the three-dimensional Helmholtz equation with variable coefficients. The coupling is achieved through a mortar variable related to an impedance trace on a smooth interface. The method has a block structure with nonsingular subblocks, and we prove the quasi-optimality of both the h and p versions of the scheme under certain conditions.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
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)
Article
Mathematics, Applied
Yanping Chen, Lina Wang, Lijun Yi
Summary: We propose an hp-discontinuous Galerkin method for solving nonlinear fractional differential equations. The method first transforms the fractional differential equations into equivalent integral equations, and then uses the hp-discontinuous Galerkin method to solve these integral equations. We prove that under certain conditions, the method can achieve exponential convergence, thus effectively solving problems with endpoint singularities.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Zhaonan Dong, Lorenzo Mascotto, Oliver J. Sutton
Summary: The novel residual-based a posteriori error estimator for the biharmonic problem in two and three dimensions gives an upper bound and a local lower bound on the error, with the lower bound being robust to local mesh size but not to local polynomial degree. The analysis is based on elliptic reconstruction and Helmholtz decomposition, showing explicit suboptimality in terms of polynomial degree that grows at most algebraically.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Dominik Schotzau, Carlo Marcati, Christoph Schwab
Summary: In this study, mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations on a polygonal subset Omega of R-2 are considered, with no-slip boundary conditions. By using corner-refined meshes and hp spaces with linearly increasing polynomial degrees, exponential rates of convergence of the method are proven for small data with piecewise analytic properties, based on recent results on analytic regularity of Leray solutions.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Maria Lukacova-Medvid'ova, Philipp Oeffner
Summary: This paper presents the convergence analysis of high-order finite element methods, with a focus on the discontinuous Galerkin scheme. By preserving structure properties and utilizing dissipative weak solutions, the convergence of the multidimensional high-order DG scheme is proven. Numerical simulations validate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
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
D. Lafontaine, E. A. Spence, J. Wunsch
Summary: The paper presents a convergence theory for the hp-FEM applied to Helmholtz problems with variable coefficients, demonstrating quasi-optimality under specific conditions and providing an upper bound on relative error. These results are important for addressing convergence issues in computational methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Engineering, Multidisciplinary
Robert E. Bird, Charles E. Augarde, William M. Coombs, Ravindra Duddu, Stefano Giani, Phuc T. Huynh, Bradley Sims
Summary: This paper presents a 2D hp-adaptive discontinuous Galerkin finite element method for phase field fracture that can reliably and efficiently solve phase field fracture problems with arbitrary initial meshes. The method uses a posteriori error estimators to drive mesh adaptivity based on both elasticity and phase field errors, and it is validated on several example problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Computer Science, Interdisciplinary Applications
Tak Shing Au Yeung, Ka Chun Cheung, Eric T. Chung, Shubin Fu, Jianliang Qian
Summary: We propose a deep learning approach to extract ray directions at discrete locations by analyzing wave fields. A deep neural network is trained to predict ray directions based on local plane-wave fields. The resulting network is then applied to solve the Helmholtz equations at higher frequencies. The numerical results demonstrate the efficiency and accuracy of the proposed scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mechanics
Sunday C. Aduloju, Timothy J. Truster
Summary: A thermodynamically consistent framework is developed for modeling ductile damage of nonlinear materials, assuming the free energy can be decomposed into elastic, plastic and damage parts. The method aims to satisfy the second law of thermodynamics, with plastic and damage dissipation localized in different domains and interfaces.
MECHANICS RESEARCH COMMUNICATIONS
(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
Computer Science, Interdisciplinary Applications
R. C. Moura, L. D. Fernandes, A. F. C. Silva, G. Mengaldo, S. J. Sherwin
Summary: In recent years, dispersion-diffusion (eigen)analyses have been used to evaluate the accuracy and stability of spectral element methods (SEMs) for under-resolved computations of transitional and turbulent flows. This study unveils a linear mechanism that causes energy transfer across Fourier modes in SEM computations and highlights the potential improvement of dissipation estimation by considering this mechanism in eigenanalysis.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
R. A. M. van Gestel, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, W. L. IJzerman
Summary: Liouville's equation describes the evolution of energy distribution in optical systems, with the discontinuous Galerkin spectral element method being suitable for this equation. Optical interfaces in phase space may lead to non-local boundary conditions which must also adhere to energy conservation. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing in terms of computational efficiency for low-error scenarios.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
