On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 52 of 188
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The Mixed Finite Element
The mixed finite element (MFE) method was developed by Raviart and Thomas
. To understand this method, consider a second order elliptic model problem
on a bounded domain Q with a Lipshitz continuous boundary F:
-V - u = f in Q (3.1)
u = Vp in Q (3.2)
p = 0 on F (3.3)
The variational form of equations (3.1) and (3.2) can be written as
Sw(V - u + f) dx = 0 Vw E L2(Q), (3.4)
jv- u dx + JpV - dx = 0 Vv E H(div; Q), (3.5)
Raviart and Thomas  proved that equations (3.4) and (3.5) have a unique
(u, p) E H(div; Q) x L2(Q). (3.6)
3.1.1 The lowest order Raviart-Thomas space
The zero order Raviart-Thomas space (RTO) and triangular elements are used in
this research work. The reason for not using higher order Raviart-Thomas space
is because of the requirement of upstream weighting, and the reason for using
triangular elements is because of the complexity of the geometrical domain.
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Deo, Milind D. On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs, report, August 31, 2005; Utah. (https://digital.library.unt.edu/ark:/67531/metadc877059/m1/52/: accessed May 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.