本文研究了稳态各向异性非均匀扩散问题的拓扑结构。讨论了两种离散化形式：a) 混合形式和 b) 直接形式。微分算子由稀疏 incidence 矩阵表示，而加权质量矩阵扮演度量依赖的 Hodge 矩阵角色。所得混合形式在右端函数 f = 0 时是点 wise 散度自由的。该方法具有 inf-sup 稳定性，并在正交和非仿射网格上显示最优收敛性。
英文摘要：
This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.
Numerical methods: mimetic finite differences and finite elements, are analyzed from a numerical point of view. It seeks to conclude on the efficiency, order of convergence and computational cost of these methods. The analysis is done in boundary value problems one-dimensional (convection-diffusion equation at steady) with different variations in the gradient, diffusion coefficient and convective velocity. Key Words: Mimetics methods, Finite Element methods, Finite differences methods, Conservative methods, Convergence.
The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences
Monash University（莫纳什大学）；Indian Institute of Technology Bombay（印度理工学院孟买分校）
In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilize the advantages of each of these methods. The usage of HMM allows us to handle anisotropic diffusion tensors on generic polygonal (polytopal) grids; whereas the CF method provides a framework for the construction of a uniformly seco…
For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $γ(\boldsymbol{n})$ in three dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important str…
Discrete maximum principle and a Delaunay-type mesh condition for linear finite element approximations of two-dimensional anisotropic diffusion problems
We solve the anisotropic diffusion equation in 2D, where the dominant direction of diffusion is defined by a vector field which does not conform to a Cartesian grid. Our method uses operator splitting to separate the diffusion perpendicular and parallel to the vector field. The slow time scale is solved using a provably stable finite difference formulation in the perpendicular to the vector field, and an integral operator for the diffusion parallel to it. Energy estimates are shown to for the continuous and semi-discrete cases. Numerical experiments are performed showing convergence of the met…
In the recent article [Kopteva, N., Numer. Math., 137, 607--642 (2017)] the author obtained residual-type a posteriori error estimates in the energy norm for singularly perturbed semilinear reaction-diffusion equations on anisotropic triangulations. The error constants in these estimates are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The purpose of this note is to improve the weights in the jump residual part of the estimator. This is attained by using a novel sharper version of the scaled trace theorem for anisotropic elements,…
Anisotropic diffusion is a well recognized tool in digital image processing, including edge detection and denoising. We present here a particular nonlinear time-dependent operator together with an appropriate high-order discretization for the space variable. The iterative procedure emphasizes the contour lines encircling the objects, paving the way to accurate reconstructions at a very low cost. One of the main features of such an approach is the possibility of relying on a rather large set of invariant discontinuous images, whose edges can be determined without introducing any approximation.
We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under {\sl anisotropic surface diffusion} with a general anisotropic surface energy $γ(\boldsymbol{n})$ in two dimensions, where $\boldsymbol{n}$ is the outward unit normal vector. By introducing a novel symmetric positive definite surface energy matrix $Z_k(\boldsymbol{n})$ depending on the Cahn-Hoffman $\boldsymbolξ$-vector and a stabilizing function $k(\boldsymbol{n})$, we first reformulate the anisotropic surface diffusion into a conservative form and then der…
Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
University of Milano–Bicocca（米兰比可卡大学）；IMATI-CNR “Enrico Magenes”（IMATI-CNR“恩里科·马盖斯”研究所）；University of Pavia（帕维亚大学）；University of Vienna（维也纳大学）；Georg-August-Universität Göttingen（哥廷根乔治-奥古斯特大学）
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffu…
We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $γ(θ)$ -- anisotropic surface diffusion -- in two dimensions, while $θ$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(θ)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized i…
We construct a new nonlinear finite volume (FV) scheme for highly anisotropic diffusion equations, that satisfies the discrete minimum-maximum principle. The construction relies on the linearized scheme satisfying less restrictive monotonicity conditions than those of an M-matrix, based on a weakly regular matrix splitting and using the Cartesian structure of the mesh (extension to quadrilateral meshes is also possible). The resulting scheme, obtained by expressing fluxes as nonlinear combinations of linear fluxes, has a larger stencil than other nonlinear positivity preserving or minimum-maxi…
Numerical resolution of an anisotropic non-linear diffusion problem
Université de Bordeaux 1, Institut de Mathématiques de Bordeaux（波尔多第一大学，波尔多数学研究所）；CNRS, Institut de Mathématiques de Toulouse（法国国家科学研究中心，图卢兹数学研究所）；CNRS, Laboratoire Paul Painlevé - Universié Sciences et Technologies de Lille（法国国家科学研究中心，保罗·潘勒韦实验室 - 里尔科学技术大学）
Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces
Université Paris-Est（巴黎东大学）；Université de Provence（普罗旺斯大学）
Anisotropic mesh adaptation for 3D anisotropic diffusion problems with application to fractured reservoir simulation
Department of Mathematics and Statistics, the University of Missouri-Kansas City（密苏里大学堪萨斯城分校数学与统计系）；Department of Mathematics, the University of Kansas（堪萨斯大学数学系）
A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields: Curvilinear coordinates and multi-block domains
Max Plank Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；The University of Texas at El Paso（得克萨斯大学埃尔帕索分校）；Proxima Fusion GmbH（Proxima Fusion有限公司）；Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学研究所）
In this paper, a new weighted first-order formulation is proposed for solving the anisotropic diffusion equations with deep neural networks. For many numerical schemes, the accurate approximation of anisotropic heat flux is crucial for the overall accuracy. In this work, the heat flux is firstly decomposed into two components along the two eigenvectors of the diffusion tensor, thus the anisotropic heat flux approximation is converted into the approximation of two isotropic components. Moreover, to handle the possible jump of the diffusion tensor across the interface, the weighted first-order f…
We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and strong anisotropies and demonstrate the capabilities of the approach for these cases numerically. The proposed model provides a variational and energy dissipative approach for anisotropic surface diffusion, enabling large scale simulations with material-specific parameters.
The anisotropic diffusion equation is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and its interplay with the ambient magnetic field. This diffusion term contributes to the highly stiff nature of the CR transport equation. In order to conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods that directly compute the exponential of the matrix to solve the linear anisotropic …
University of Tehran（德黑兰大学）；Shahid Beheshti University（沙希德贝赫什提大学）；Institute for Cognitive and Brain Sciences（认知与脑科学研究所）；Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）
Anisotropic diffusion filtering for signal smoothing as a low-pass filter has the advantage of the edge-preserving, i.e., it does not affect the edges that contain more critical data than the other parts of the signal. In this paper, we present a numerical algorithm based on least squares support vector regression by using Legendre orthogonal kernel with the discretization of the nonlinear diffusion problem in time by the Crank-Nicolson method. This method transforms the signal smoothing process into solving an optimization problem that can be solved by efficient numerical algorithms. In the f…
University of Cambridge（剑桥大学）；University of Warwick（华威大学）；Austrian Academy of Sciences（奥地利科学院）
In many problems in data classification one wishes to assign labels to points in a point cloud with a certain number of them being already correctly labeled. In this paper, we propose a microscopic ODE approach, in which information about correct labels is propagated to neighboring points. Its dynamics are based on alignment mechanisms, which are commonly used in large interacting agent systems in consensus formation. We derive the respective continuum description, which corresponds to an anisotropic diffusion equation with reaction term. Solutions of the continuum model on the bounded domain …
We introduce a variant of the Gatenby-Gawlinski model for acid-mediated tumor invasion, accounting for anisotropic and heterogeneous diffusion of the lactic acid across the surrounding healthy tissues. Numerical simulations are performed for two-dimensional data by employing finite volume schemes on staggered Cartesian grids, and parallel implementation through the modern CUDA GPUs technology is considered. The effectiveness of such approach is proven by reproducing biologically relevant results like the formation of propagating fronts and the emergence of an interstitial gap between normal an…
