Navert, U., A finite element method for convection-diffusion problems, Thesis, Chalmers University of Technology and University of Gothenburg 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....diffusion method is a finite element method introduced by Hughes and Brooks [2] in the context of stationary convection diffusion problems. Mathematical analyses of the method have been performed by Johnson and Navert [3] Johnson et al. 5] and Niijima [8] for stationary problems. Navert [7] extended the method to time dependent convection diffusion problems and obtained local L 2 error estimates of order k 1=2, for piecewise polynomial finite elements of degree k, in smooth regions (i.e. regions away from any layers) Many other authors have applied the streamline diffusion ....

....such that 0 S j Psi ; 2.9) 3 where C(S j ) denotes the space of continuous functions on S j and 0 is the interior of . We also introduce the streamline derivative w fi for all differentiable functions w by defining w fi = aw x w t : We shall apply the streamline diffusion method [7, 4] to the problem (1.1) on each slab S j successively, imposing the initial value at t = t j Gamma1 weakly and the boundary condition strongly. To this end, we introduce the finite element space on Omega Gamma V j n v 2 L 2( Omega Gamma : v fi fi S j 2 V j for j = 1; M o ; ....

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Navert, U., A finite element method for convection-diffusion problems, Thesis, Chalmers University of Technology and University of Gothenburg 1982.

....artificial diffusion only along the streamline direction to suppress possible oscillations near discontinuities or sharp layers of the solution without causing overly large crosswind diffusion. Mathematical studies of the method were started by Johnson and Navert [46] and continued by Navert [61] and Johnson et al. 47, 50] The work includes the extension to the time dependent problem with the discontinuous Galerkin methodology. Primary stability, convergence and crosswind smear evaluation results were established for the model problem. The SD method gives added stability as compared ....

U. Navert, A finite Element Method for Convection-Diffusion Problems. Thesis, Chalmers Univ. of Tech. and Univ. of Gothenburg, (1982).

....L 2( Omega Gamma 8v 2 V h in which the lower bound is positive in the limit 0. In particular, the finite element matrix has positive definite symmetric part and the discrete solution u h of (3) is unique. If u is the strong solution and f 2 L 2( Omega Gamma3 then Axelsson [1] and Navert [18] have shown that for a piecewise linear finite element space there is a constant C (independent of h; ffi s and ) such that 1=2 kr(u Gamma u h )k ffi 1=2 s k(u Gamma u h ) fi k C i 1=2 h ffi 1=2 s h j juj 2 ; where k Delta k and j Delta j 2 denote the usual L 2 ....

U. Navert. A Finite Element Method for Convection-Diffusion Problems. PhD thesis, Chalmers University of Technology, 1982.

....solution converges uniformly if in some norm the estimate ku Gamma u h k Ch p (0.3) email: marc mathematik.uni freiburg.de holds uniformly in . The constant should also not depend implicitly on , e.g. depend on some norm of the exact solution as for the streamline diffusion method [Nav82] or the upwind finite volume method [KR96] For uniformly convergent discretisations we refer to [Il 69] MW78] Gar88b] HMMR95] and [RST96] A discretisation of (0.1) results in the problem of solving a large and sparse linear system. Our aim is to construct a robust twogrid method for ....

U. Navert. A Finite Element Method for Convection-Diffusion Problems. PhD thesis, Chalmers University, Goteborg, Sweden, 1982.

....element method (SDFEM) for the solution of convection diffusion problems has been successfully applied in the case of conforming finite element spaces. This method was proposed first by Hughes and Brooks [6] and applied to several classes of problems. Starting with the fundamental work by Navert [13], it was mainly analyzed by Johnson and his co workers [5,9,10] Nowadays, the convergence properties are well understood in the conforming case [14,17,21,22,24] However, in the nonconforming case there are some new effects which will be studied in this paper. For applications from computational ....

U. Navert. A finite element method for convection-diffusion problems. PhD thesis, Chalmers University of Technology Goteborg, 1982.

....; and a set fffi e g of non negative numerical diffusion parameters. Here, G denotes the inner product in L 2 (G) G Omega Gamma In view of the difficulties to get a priori information on the solution u we restrict our consideration to a certain class of problems introduced in Navert [8] which allows to localize the boundary layers R of thickness b Figure 1: Anisotropic mesh in the boundary layer region O( ln 1 ) 1 2 or = 1, at some straight lines M ae Then, we assume the following hypothesis is to be satisfied: kD ff j; u; L 2 (e)k K(f) q meas(e) ....

U. Navert. A finite element method for convection-diffusion problems. PhD thesis, Chalmers University of Technology, Goteborg, 1982.

....dist(x; T j Figure 4. 1: Anisotropic mesh in the boundary layer region O( p ln 1 ) are located at Omega with the exception of the inflow boundary part at x 1 = 1 (no layer) and the outflow boundary part at x 1 = 0 where again ordinary boundary layers of thickness O( ln 1 ) occur [21]. In Section 5 we consider a more general type of domain, but only in the two dimensional case. 4.2 Mesh generation with anisotropic boundary layer refinement The idea is now ffl to construct a fixed mesh in the boundary layer region with anisotropic refinement and ffl to use an isotropic mesh ....

U. Navert. A finite element method for convection-diffusion problems. PhD thesis, Chalmers University of Technology, Goteborg, 1982.

....jjjvjjj 2 sd (9) in which the lower bound is positive in the limit 0. In particular, the finite element discretization matrix has positive definite symmetric part and the discrete solution u h of (3) is unique. If f 2 L 2 (W) and u is the strong solution, then Axelsson [1] and N avert [28] have 1 If Th is not uniform or fi is a variable, then let h represent the diameter of a local element, and determine ffi s elementwise as in (6) see [22, p. 186] shown that for a piecewise linear finite element space there is a constant C (independent of h; ffi s and ) such that jjju ....

U. Navert. A Finite Element Method for Convection-Diffusion Problems. PhD thesis, Chalmers University of Technology, 1982.

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