We study a two-scale reaction-diffusion system with nonlinear reaction terms and a nonlinear transmission condition (remotely ressembling Henry's law) posed at air-liquid interfaces. We prove the rate of convergence of the two-scale Galerkin method proposed in Muntean & Neuss-Radu (2009) for approximating this system in the case when both the microstructure and macroscopic domain are two-dimensional. The main difficulty is created by the presence of a boundary nonlinear term entering the transmission condition. Besides using the particular two-scale structure of the system, the ingredients of the proof include two-scale interpolation-error estimates, an interpolation-trace inequality, and improved regularity estimates.
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks. The finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful…
We study the order of convergence of Galerkin variational integrators for ordinary differential equations. Galerkin variational integrators approximate a variational (Lagrangian) problem by restricting the space of curves to the set of polynomials of degree at most $s$ and approximating the action integral using a quadrature rule. We show that, if the quadrature rule is sufficiently accurate, the order of the integrators thus obtained is $2s$.
Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions
In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the time-dependent Oseen equation. The generalized weak Galerkin method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical scheme are developed and analyzed for their convergence, stability, and error estimates. A generalized {\em{inf-sup}} condition is developed to assist the convergence analysis. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are establi…
An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries
Beijing Computational Science Research Center（北京计算科学研究中心）；School of Mathematical Sciences, Guizhou Normal University（贵州师范大学数学科学学院）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
This study presents a two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC). Firstly, the experimental order of convergence of the scheme is shown to see the second-order accuracy in time. Secondly, the effect of the TBC on a simple domain is discussed; the artificial reflections are kept from the Dirichlet boundaries and removed significantly from the transmission boundaries. Thirdly, the scheme is applied to a complex practical domain, i.e., the Bay of Bengal region, which is non-convex and includes islands. The effect of the TBC is di…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of deterministic convection diffusion equations, is used to handle the stochastic domain in this study, whereas discontinuous Galerkin method is used to discretize spatial domain due to its local mass conservativity. A priori error estimates of the stationary problem and stability estimate of the unsteady model problem are derived in the energy norm. To address the c…
Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems
Computer, Electrical, Mathematical Sciences and Engineering Division, KAUST（计算机、电气、数学科学与工程系，阿卜杜拉国王科技大学）；Department of Mathematics, University of Sussex（萨塞克斯大学数学系）；Institute of Applied & Computational Mathematics, Foundation for Research & Technology-Hellas（应用与计算数学研究所，希腊研究与技术基金会）；Department of Mathematics and Applied Mathematics, University of Crete（克里特大学数学与应用数学系）；Faculty of Civil Engineering-Czech Technical University of Prague（捷克布拉格工业大学土木工程学院）
In this paper we study the numerical method for approximating the random periodic solution of semiliear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well-defined in the intersection of a family of decreasing Hilbert spaces. Then we propose a Galerkin-type exponential integrator scheme and establish its conve…
We consider two `Classical' Boussinesq type systems modelling two-way propagation of long surface waves in a finite channel with variable bottom topography. Both systems are derived from the 1-d Serre-Green-Naghdi (SGN) system; one of them is valid for stronger bottom variations, and coincides with Peregrine's system, and the other is valid for smaller bottom variations. We discretize in the spatial variable simple initial-boundary-value problems (ibvp's) for both systems using standard Galerkin-finite element methods and prove $L^2$ error estimates for the ensuing semidiscrete approximations.…
We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In thi…
Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations
The University of Tennessee（田纳西大学）；University of Pittsburgh（匹兹堡大学）
We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver $O(h^{k+1})$ $H^2$-error estimates when using polynomials of order $k\ge0$ to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld--Schöberl for second-order PDEs. The cell unknowns are polynomials of order $(k+2)$ that can be eliminated locally by means of static condensation. The face unknowns approximati…
Symplectic Hamiltonian Direct Discontinuous Galerkin Method for Wave Propagation
Sino-French Carbon Neutrality Research Center, Centrale Pekin, Beihang University（中法碳中和研究中心，北京中央大学，北京航空航天大学）；LMIB and School of Mathematical Sciences, Beihang University（数学建模与仿真实验室及数学科学学院，北京航空航天大学）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, National Center for Applied Mathematics in Hunan, Xiangtan University（湖南省科学与工程计算重点实验室，湖南应用数学中心，湘潭大学）；International Research Center for Mathematics and Interdisciplinary Sciences, Hangzhou International Innovation Institute of Beihang University（数学与交叉科学国际研究中心，北京航空航天大学杭州国际创新研究院）
Error Estimates of the Gain Approximation by Hermite-Galerkin Method in Feedback Particle Filter
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；Key Laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University（北京航空航天大学数学、信息与行为语义重点实验室）
Very high order treatment of embedded curved boundaries in compressible flows: ADER discontinuous Galerkin with a space-time Reconstruction for Off-site data
针对非贴合网格上弯曲边界的高阶边界条件一致性难题，将 ROD 融入 ADER 间断 Galerkin 的时空预测器，实现单次时空多项式重构，并通过后验限制支持激波与弯曲边界相互作用计算。
