The diffusion problem in mixed form
Chapter
Abstract
The diffusion problem in a mixed form is governed by the following set of equations:
where the vector variable u represents the flux of the scalar unknown p. The unknown p may be a pressure, a temperature, or a flow density depending on the physical interpretation that we give to this mathematical model. The mixed form of the diffusion problem provides an opportunity for a better approximation of the flux and the exact satisfaction of balance condition (5.2), e.g., [ 90, 205, 208].
$${\mathbf{u}} + \user1{K}\nabla p = 0\;\;\;\;\;\;\;{\text{in}}\;\Omega ,$$
(5.1)
$${\text{div}}\,{\mathbf{u}} = b\;\;\;\;\;\;\;{\text{in}}\;\Omega ,$$
(5.2)
$$p = {g^D}\;\;\;\;\;\;\;{\text{on}}\;{\Gamma ^D},$$
(5.3)
$${\mathbf{u}} \cdot {\mathbf{n}} = - {g^N}\;\;\;\;{\text{on}}\;{\Gamma ^N},$$
(5.4)
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