This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

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Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that th...

Basic principles of Virtual Element Methods

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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Finite elements in computational electromagnetism

We present an algorithm to efficiently and robustly process collisions, contact and friction in cloth simulation. It works with any technique for simulating the internal dynamics of the cloth, and allows true modeling of cloth thickness. We also show how our simulation data can be post-processed with a collision-aware subdivision scheme to produce smooth and interference free data for rendering.

/pdf/robust-treatment-of-collisions-contact-and-friction-for-2ngobdlzjz.pdf

Robust treatment of collisions, contact and friction for cloth animation

http://www.math.pitt.edu/~yotov/teaching/16-2/math2602/MFD-JCP-survey.pdf

Mimetic finite difference method

http://cnls.lanl.gov/~shashkov/papers/comp.pdf

The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy

The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

/pdf/convergence-of-the-mimetic-finite-difference-method-for-5gf821ts3b.pdf

Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes

INTRODUCTION Governing Equations Elliptic Equations Heat Equation Equation of Gas Dynamic in Lagrangian Form The Main Ideas of Finite-Difference Algorithms 1-D Case 2-D Case Methods of Solution of Systems of Linear Algebraic Equation Methods of Solution of Systems of Nonlinear Equations METHOD OF SUPPORT-OPERATORS Main Stages The Elliptic Equations Gas Dynamic Equations System of Consistent Difference Operators in 1-D Inner Product in Spaces of Difference Functions and Properties of Difference Operators System of Consistent Difference Operators in 2-D THE ELLIPTIC EQUATIONS Introduction Continuum Elliptic Problems with Dirichlet Boundary Conditions Continuum Elliptic Problems with Robin Boundary Conditions One-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Centered Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Two-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Valued Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Conclusion Two-Dimensional Support Operator Algorithms Discretization Spaces of Discrete Functions The Prime Operator The Derived Operator Multiplication by a Matrix and the Operator D The Difference Scheme for the Elliptic Operator The Matrix Problem Approximation and Convergence Properties HEAT EQUATION Introduction Finite-Difference Schemes for Heat Equation in 1-D Finite-Difference Schemes for Heat Equation in 2-D LAGRANGIAN GAS DYNAMICS Kinematics of Fluid Motions Integral Form of Gas Dynamics Equations Integral Equations for One Dimensional Case Differential Equations of Gas Dynamics in Lagrangian Form The Differential Equations in 1D. Lagrange Mass Variables The Statements of Gas Dynamics Problems in Lagrange Variables Different Forms of Energy Equation Acoustic Equations Reference Information Characteristic Form of Gas Dynamics Equations Riemann's Invariants Discontinuous Solutions Conservation Laws and Properties of First Order Invariant Operators Finite-Difference Algorithm in 1D Discretization in 1D Discrete Operators in 1D Semi-Discrete Finite-Difference Scheme in 1D Fully Discrete, Explicit, Computational Algorithm Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Stability Conditions Homogeneous Finite-Difference Schemes. Artificial Viscosity Artificial Viscosity in 1D Numerical Example Finite Difference Algorithm in 2D Discretization in 2D Discrete Operators in 2D Semi-Discrete Finite-Difference Scheme in 2D Stability Conditions Finite-Difference Algorithm in 2D Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Artificial Viscosity in 2D Numerical Example APPENDIX: FORTRAN CODE DIRECTORY General Description of Structure of Directories on the Disk Programs for Elliptic Equations Programs for 1D Equations Programs for 2D Equations Programs for Heat Equations Programs for 1D Equations Programs for 2D Equations Programs for Gas Dynamics Equations Programs for 1D Equations Programs for 2D Equations Bibliography

Conservative Finite-Difference Methods on General Grids

https://cnls.lanl.gov/~shashkov/papers/ed_vis.pdf

Mikhail Shashkov

Papers

Mimetic finite difference method

The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy

Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes

Conservative Finite-Difference Methods on General Grids

Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations