Yotov, I.: Mixed finite element methods for flow in porous media, PhD thesis, Rice University, Houston, Texas, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....nal equation enforces continuity of the hybrid mixed method side pressures along the coarse ne interface; its nite di erence form can be found in equation (74) below. The matrices R f and P c depend on the choice of the mortar elements, and will be described in section 7.1.2 below. It is known [46, 92] that the choice of mortar elements h produces a stable approximation for which the error estimate is satis ed. If the regularity of the problem would allow, a higher order error estimate would be available, in which the pressure would be second order accurate and the velocities would have order ....

Ivan Yotov. Mixed Finite Element Methods for Flow in Porous Media. PhD thesis, Rice University, Texas, 1996. 72

.... near coarse fine mesh interfaces [31, 38, 58] Recent developments in the application of the mixed finite element method to more general mesh geometries and locally refined grids have incorporated Lagrange multipliers along mesh interfaces to improve flux approximation near such interfaces [83]. We have not yet considered the latest approaches to preserve flux continuity on composite meshes nor approaches to handle full tensor permeability fields (e.g. control volume finite element methods [47] although this will be a subject of future work. For random permeability fields, or ....

I. Yotov. Mixed Finite Element Methods for Flow in Porous Media. PhD thesis, The University of Texas at Austin, 1996. 34

....to capture the spatial behavior of the solution. In that case, non overlapping domain decomposition techniques with Mortar elements at the interfaces of the decomposition have proven to be e#cient since they enable to define the grids independently in the subdomains regions (see [GW88] [Yot96]) ACWY96] On the other hand, the transient behavior of the solution may also warrant the use of di#erent time steps in the di#erent subdomains. The idea of the domain decomposition method introduced in this paper is to combine Mortar Mixed Finite Element methods for the space discretization ....

....i )d# #v#V , and we shall denote by H 1 2 (#) the subspace of L 2 (#) of functions such that ## 1 2 ,# #. We consider, on the domain decomposition(# i ) i=1, N , a Mortar Mixed Finite Element (MMFE) discretization of (1) introduced in [GW88] for matching grids, and extended in [Yot96], ACWY96] to the case of non matching grids at the interfaces between the subdomains# i . In that case, a so called Mortar space # h # L 2 (#) is introduced on the skeleton #. Then, equation (1) is discretized on each subdomain by a Mixed Finite Element Method, and the matching at the ....

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I Yotov. Mixed Finite Element Methods for Flow in Porous Media. PhD thesis, TICAM, University of Texas at Austin, 1996.

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I. Yotov, Mixed finite element methods for flow in porous media, Ph.D. thesis, Rice University, Houston, TX (1996); also TR96-09, Dept. Comp. Appl. Math., Rice University and TICAM Report 96-23, University of Texas at Austin.

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YOTOV, I., Mixed Finite-Element Methods for Flow in Porous Media, PhD Thesis, Rice University, 1996.

....obtain the same order of convergence in both cases. The method using discontinuous mortars provides better local mass conservation across the interfaces, but numerical observations suggest that this may lead to slightly bigger numerical error. The method presented here has also been considered in [27] in the case of the lowest order Raviart Thomas spaces [23, 22] Here we take a somewhat different approach in the analysis, which allows us to relax a condition on the mortar grids needed to obtain optimal convergence and superconvergence. The relaxed condition is easily satisfied in practice. An ....

....OE 2 h , d h;i (OE; OE) is equivalent to jI Omega i Q h;i OEj 1=2; Omega i , where I Omega i is an interpolation operator onto the space of continuous piece wise linears on Omega i . Therefore k Delta k dh can be characterized as a certain discrete H 1=2 norm on Gamma (see [27]) This is also in accordance with the numerically observed O(h 2 ) convergence for the mortars in a discrete L 2 norm (see Section 8) MIXED METHODS ON NON MATCHING GRIDS 15 7. A substructuring domain decomposition algorithm. In this section we discuss the implementation of a parallel ....

I. Yotov, Mixed finite element methods for flow in porous media, PhD thesis, Rice University, Houston, Texas, 1996.

....obtain the same order of convergence in both cases. The method using discontinuous mortars provides better local mass conservation across the interfaces, but numerical observations suggest that this may lead to slightly bigger numerical error. The method presented here has also been considered in [30] in the case of the lowest order Raviart Thomas spaces [26, 25] Here we take a somewhat different approach in the analysis, which allows us to relax a condition on the mortar grids needed to obtain optimal convergence and superconvergence. The relaxed condition is easily satisfied in practice. An ....

....3.1. Condition (3.18) implies the solvability condition (2.14) which is simply (3.18) wherein we allow C to vary with h. So (3.18) strengthens (2.14) so that it holds uniformly as h tends to zero. This is not a very restrictive condition, and it is easily satisfied in practice. It can be shown [30] that (3.18) holds for both continuous and discontinuous mortar spaces, if the the mortar grid on each interface is a coarsening by two in each direction of the trace of either one of the subdomain grids. This choice is reminiscent of the one in the case of standard or spectral finite element ....

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I. Yotov, Mixed finite element methods for flow in porous media, PhD Thesis, Rice University, Houston, Texas, 1996.

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Yotov, I.: Mixed finite element methods for flow in porous media, PhD thesis, Rice University, Houston, Texas, 1996.

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