We study two techniques for correcting the geometrical error associated with domain approximation by a polygon. The first was introduced some time ago \cite{bramble1972projection} and leads to a nonsymmetric formulation for Poisson's equation. We introduce a new technique that yields a symmetric formulation and has similar performance. We compare both methods on a simple test problem.
Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding solutions. While recent works have analyzed this issue for a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal…
A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
Université de Lille 1（里尔第一大学）；INRIA Lille Nord Europe（法国国家信息与自动化研究所里尔北部分支）；Université de Valenciennes et du Hainaut Cambrésis（瓦朗谢纳及康布雷埃斯海诺特大学）
We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan (H. Liu, J. Yan, The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections, Communications in Computational Physics 8 (3) (2010) 541) to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresp…
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indica…
This work analyzes a high order hybridizable discontinuous Galerkin (HDG) method for the linear elasticity problem in a domain not necessarily polyhedral. The domain is approximated by a polyhedral computational domain where the HDG solution can be computed. The introduction of the rotation as one of the unknowns allows us to use the gradient of the displacements to obtain an explicit representation of the boundary data in the computational domain. The boundary data is transferred from the true boundary to the computational boundary by line integrals, where the integrand depends on the Cauchy …
Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived …
Isogeometric approach applied to Boundary Element Methods is an emerging research area. In this context, the aim of the present contribution is that of investigating, from a numerical point of view, the Symmetric Galerkin Boundary Element Method (SGBEM) devoted to the solution of 2D boundary value problems for the Laplace equation, where the boundary and the unknowns on it are both represented by B-splines. We mainly compare this approach, which we call IGA-SGBEM, with a curvilinear SGBEM, which operates on any boundary given by explicit parametric representation and where the approximate solu…
A Trefftz Discontinuous Galerkin Method for Time Harmonic Waves with Generalized Impedance Boundary Conditions
Institute for Mathematics and its Applications, College of Science and Engineering（数学及其应用研究所，科学与工程学院）；Department of Mathematical Sciences, University of Delaware（特拉华大学数学科学系）；Departamento de Matemáticas, Universidad de Oviedo（奥维耶多大学数学系）
Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potential…
We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficien…
Spectral-Galerkin Approximation and Optimal Error Estimate for Stokes Eigenvalue Problems in Polar 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）；State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所计算机科学与技术国家重点实验室/并行计算实验室）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
A strong second-order two-stage explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for two-dimensional Gray-Scott model
Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊玛目穆罕默德·伊本·沙特伊斯兰大学）
Discontinuous Galerkin Isogeometric Analysis on Non-matching Segmentation: Error Estimates and Efficient Solvers
Johannes Kepler University (JKU)（约翰内斯·开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院计算与应用数学约翰·拉德恩研究所）
The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes.The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the $L^2$ and energy norm. We assess the effec…
This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $α>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which …
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）
We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform…
Minimization-based polynomial corrections for high-order curved boundaries on fixed and moving domains: assessment on finite volume and discontinuous Galerkin schemes
Université Paris Cité and Sorbonne Université, CNRS, LJLL（巴黎西岱大学与索邦大学，法国国家科学研究中心，拉格朗日分析与应用数学实验室）；Laboratoire de Mathématiques, Université Savoie Mont Blanc, CNRS, UMR 5127（萨瓦-蒙布朗大学数学实验室，法国国家科学研究中心）；Department of Mathematics and Computer Science, University of Ferrara（费拉拉大学数学与计算机科学系）
A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
Adaptive spectral (AS) decompositions associated with a piecewise constant function $u$ yield small subspaces where the characteristic functions comprising $u$ are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive $L^2$-error estimates for the AS decomposition of $u$, truncated after $K$ terms, when $u$ is piecewise constant and consists of $K$ characteristic functions over Lipschitz domains…
We present a stabilized POD-Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the su…
Ray flow methods are an efficient tool to estimate vibro-acoustic or electromagnetic energy transport in complex domains at high-frequencies. Here, a Petrov-Galerkin discretization of a phase-space boundary integral equation for transporting wave energy densities on two-dimensional surfaces is proposed. The directional dependence of the energy density is approximated at each point on the boundary in terms of a finite local set of directions propagating into the domain. The direction of propagation can be preserved for transport across multi-component domains when the directions within the loca…
