Finite Element Approximation of the Diffusion Operator on Tetrahedra

Putti, Mario; Cordes, C.

doi:10.1137/s1064827595290711

Cited by 57 publications

(76 citation statements)

References 13 publications

(25 reference statements)

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“…Having applied the viscous term discretization to the Laplace operator, Delanaye et al [8] showed that a correction of the diamond-shaped control volume on Cartesian grids leads to the more positive scheme and obtained a robust discretization of the viscous terms in Navier-Stokes equations. Putti and Cordes [13] have proposed a modification of the control volume that allowed them to obtain the positive discretization of the Laplace equation on three-dimensional Delaunay meshes. In all these cases the choice of the control volume has been dictated by the geometry of grid cells.…”

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confidence: 99%

Modification of A Finite Volume Scheme for Laplace's Equation

Petrovskaya¹

2001

SIAM J. Sci. Comput.

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Abstract. For Laplace's equation, we discuss whether it is possible to construct a linear positive finite volume (FV) scheme on arbitrary unstructured grids. Dealing with the arbitrary grids, we state a control volume which guarantees a positive FV scheme with linear reconstruction of the solution. The control volume is defined by a property of the analytical solution to the equation and does not depend on the grid geometry. For those problems where the choice of the control volume is prescribed a priori, we demonstrate how to improve positivity of the linear FV scheme by using corrected reconstruction stencils. The difficulties arising when grids with no geometric restrictions are used for the discretization are discussed. Numerical examples illustrating the developed approach to the stencil correction are given.Key words. Laplace's equation, finite volume scheme, positivity, stencil correction AMS subject classifications. 65N12, 74S10, 35J05PII. S1064827500368925 Introduction.A discrete Laplace operator is often considered to be a good model for investigating a discretization of partial differential equations which contain diffusion operator ∇·(D∇). Two important examples are given by convection-diffusion equations and Navier-Stokes equations with possible applications that include the problems of fluid dynamics, chemical engineering, and environmental pollution. For the numerical solution of these equations, what is desirable are discretization schemes which satisfy a discrete maximum principle (monotone schemes); otherwise one can expect strong oscillations or even divergency of the solution.There are two possible approaches for development of monotone schemes on unstructured grids. The first approach is to use the grids with some geometric constraints on the triangulation. It is well known that triangulations with no obtuse triangles allow us to construct monotone schemes. The relevant examples are given in [6,11]. However, nonobtuse triangulations can be used on very few practical problems. In the process of the grid generation it is often required to resolve some complicated features of the problem geometry that makes strict angle control to be difficult.Looking for a wider class of triangulations, in the two-dimensional case a Delaunay triangulation [10] is very attractive. For Laplace's equation, in the two-dimensional case the Delaunay triangulation provides positivity (that guarantees the discrete maximum principle) of the linear finite element/finite volume scheme (Barth [3]). This important property may be applied for the solution of a wide range of problems, even more general than discretization of the Laplace equation. For instance, Xu and Zikatanov [17] developed a linear monotone finite element scheme for convectiondiffusion equations in any spatial dimension. To obtain a positive discretization of the diffusion operator they assumed the restrictive geometrical conditions which in the two-dimensional case mean that the triangulation is a Delaunay triangulation.

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Modification of A Finite Volume Scheme for Laplace's Equation

Petrovskaya¹

2001

SIAM J. Sci. Comput.

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“…N T and N represent the total number of triangles and nodes (or vertices) of T h , respectively, and T e , e = 1, …, N T , is a generic triangle of T h , with area We construct a dual mesh over T h , where a dual finite volume e i , i =1, ..., N, is associated with node i. It is the Voronoi polygon, defined as in [40], shown in Figure 2b. We call the dual finite volumes, (computational) cells.…”

Section: Computational Mesh Propertiesmentioning

confidence: 99%

Simulation of the Propagation of Tsunamis in Coastal Regions by a Two-Dimensional Non-Hydrostatic Shallow Water Solver

Aricò¹

2017

FMRIJ

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During the last years, great effort has been addressed by several authors to simulate the propagation of solitary waves/ tsunamis, tides or surges, due to the tremendous damages and losses of human lives in the inundated rural and residential areas. Tsunamis are sea waves usually generated by undersea landslides and earthquakes. They can be regarded as long/solitary waves with small amplitude and long wavelength, travelling with high speed over long distances. Approaching the coast, their amplitude increases, becoming potentially destructive. The propagation of tsunami in coastal regions can be studied by investigating the shoaling and breaking of solitary waves over inclined bottoms [1]. Chanson [2] asserted that the front of tsunamis over dry plains becomes a shock wave, and presented a similarity between the propagation of the tsunamis over dry coastal areas and the classical dam-break problems. Generally, 3D simulations of the processes described as above, e.g., by RANS models [3], or Smoothed Particle Hydrodynamics methods [4], or Volume of Fluids methods [5], require very high computational costs. For these reasons, a significant amount of literature based on depthintegrated equations has been published for simulations of long waves/tsunamis propagation. Generally, the two modeling approaches proposed in literature are the Boussinesq-Type Models (BTMs) and the Non-Hydrostatic Nonlinear Shallow Water Equations (NLSWEs) models.The classical formulation of the BTMs [6] involves weak nonlinearity and dispersion. High-order BTMs have been proposed to overcome these difficulties, but suffer from complex numerical discretization, which leads to high computational efforts [7,8]. Generally, the BTMs also suffer from the inclusion, in the governing equations, of extra terms, based on empirical considerations, introduced for the simulation of wave breaking with the associated energy dissipation, and wetting/drying transition [8].Hereafter, we call "hydrostatic NLSWEs" and "non-hydrostatic NLSWEs" (or "dynamic NLSWEs") respectively the NLSWEs with hydrostatic and non-hydrostatic (or dynamic) pressure distribution. Due to simplicity and accuracy, the hydrostatic AbstractDue to the enormous damages and losses of human lives in the inundated regions, the simulation of the propagation of tsunamis in coastal areas has received an increasing interest of the researchers. We present a 2D depth-integrated, nonhydrostatic shallow waters solver to simulate the propagation of tsunamis, solitary waves and surges in coastal regions. We write the governing continuity and momentum equations in conservative form and discretize the domain with unstructured triangular Generalized Delaunay meshes. We apply a fractionaltime-step procedure, where two problems (steps) are consecutively solved. In the first and in the second step, we hypothesize a hydrostatic and a non-hydrostatic distribution of the pressure, respectively. Several literature models, which solve the same set of equations, are based on a fractional-time-step procedure. In th...

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“…and h km is a water depth value in the computational cell obtained by local mass balance, computed as explained in [14], as well as the Q sp coefficient in Equation (20). The initial condition is η = 0, and L(h km ) in Equation (20) is the water surface width corresponding to h km .…”

Section: The Fractional Time Step Proceduresmentioning

confidence: 99%

“…The dual finite volume e i , previously defined, is called the Voronoi cell or the Thiessen polygon [20]. The vertices of the Voronoi cells are the circumcenters of the Delaunay triangulation.…”

Section: Computational Mesh Properties and Computational Cellsmentioning

confidence: 99%

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The FLO Diffusive 1D-2D Model for Simulation of River Flooding

Aricò

Filianoti

Sinagra

et al. 2016

Water

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An integrated 1D-2D model for the solution of the diffusive approximation of the shallow water equations, named FLO, is proposed in the present paper. Governing equations are solved using the MArching in Space and Time (MAST) approach. The 2D floodplain domain is discretized using a triangular mesh, and standard river sections are used for modeling 1D flow inside the section width occurring with low or standard discharges. 1D elements, inside the 1D domain, are quadrilaterals bounded by the trace of two consecutive sections and by the sides connecting their extreme points. The water level is assumed to vary linearly inside each quadrilateral along the flow direction, but to remain constant along the direction normal to the flow. The computational cell can share zero, one or two nodes with triangles of the 2D domain when lateral coupling occurs and more than two nodes in the case of frontal coupling, if the corresponding section is at one end of the 1D channel. No boundary condition at the transition between the 1D-2D domain has to be solved, and no additional variable has to be introduced. Discontinuities arising between 1D and 2D domains at 1D sections with a top width smaller than the trace of the section are properly solved without any special restriction on the time step.

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Finite Element Approximation of the Diffusion Operator on Tetrahedra

Cited by 57 publications

References 13 publications

Modification of A Finite Volume Scheme for Laplace's Equation

Modification of A Finite Volume Scheme for Laplace's Equation

Simulation of the Propagation of Tsunamis in Coastal Regions by a Two-Dimensional Non-Hydrostatic Shallow Water Solver

The FLO Diffusive 1D-2D Model for Simulation of River Flooding

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