scholarly journals Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case

2017 ◽  
Vol 9 (1) ◽  
pp. 68 ◽  
Author(s):  
Michel Fortin ◽  
Abdellatif Serghini Mounim
Keyword(s):  
Finite Volume ◽  
Numerical Approximation ◽  
Dimensional Case ◽  
Parabolic Problems ◽  
Finite Volume Schemes ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Hybrid Finite Element ◽  
The One ◽  
Diffusion Problems

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

scholarly journals Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for Transport

2007 ◽  
Vol 17 (3) ◽  
pp. 321-328 ◽  
Author(s):  
Bruno Després ◽  
Frédéric Lagoutière
Keyword(s):  
Finite Volume ◽  
Transverse Direction ◽  
Transport Equations ◽  
Discrete Solution ◽  
Finite Volume Schemes ◽  
Time Step ◽  
One Dimensional ◽  
Transport Velocity ◽  
Volume Algorithm ◽  
The One

Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for TransportWe develop a new multidimensional finite-volume algorithm for transport equations. This algorithm is both stable and non-dissipative. It is based on a reconstruction of the discrete solution inside each cell at every time step. The proposed reconstruction, which is genuinely multidimensional, allows recovering sharp profiles in both the direction of the transport velocity and the transverse direction. It constitutes an extension of the one-dimensional reconstructions analyzed in (Lagoutière, 2005; Lagoutière, 2006).

Error estimates for higher-order finite volume schemes for convection–diffusion problems

2018 ◽  
Vol 26 (1) ◽  
pp. 35-62
Author(s):  
Dietmar Kröner ◽  
Mirko Rokyta
Keyword(s):  
Error Estimates ◽  
Finite Volume ◽  
Original Problem ◽  
A Priori ◽  
Higher Order ◽  
Finite Volume Schemes ◽  
Convection Diffusion ◽  
Type Reconstruction ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIt is still an open problem to provea priorierror estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domainΩin ℝ2and we can prove such kind of ana priorierror estimate. In the part of the estimate, which refers to the discretization of the convective term, we gainh1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.

scholarly journals Finite volume discretisation for the one-dimensional convection diffusion-dissipation equation

2016 ◽  
Vol 5 (4) ◽  
pp. 206
Author(s):  
Bienvenu ONDAMI
Keyword(s):  
Finite Volume ◽  
Water Table ◽  
Method Of Lines ◽  
Finite Volume Scheme ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Volume Scheme ◽  
The One ◽  
Dissipation Equation

This paper is devoted to analysis of a finite volume scheme for a one-dimensional convection-diffusion-dissipation equation having application in pollution of water table. We analyse a scheme corresponding to a semi-descretization, also called method of lines. Results of umerical experiments using this approach are reported.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

Sign in / Sign up

Export Citation Format

Share Document