Abstract
This work is devoted to a modified fixed point method applied to the bio-chemical transport equation. To have a good accuracy for the solution we treat, we apply an implicit scheme to this equation and use a modified fixed point technique to linearize the problem of transport equation with a generalized nonlinear reaction and diffusion equation. Next, we apply this methods in particular to the the dynamical system of a bio-chemical process. Eventually, we accelerate these algorithms by the optimized domain decomposition methods.Several test-cases of analytical problems illustrate this approach and show the efficiency of the proposednew method.
Highlights
Of finite element or finite volume discretization is sensitive to the anisotropic and convective equations: modifying equations by q relaxation procedure solves this issue and we show the weak formulation of problem (1.1), has one and unique solution
H is the specific Sobolev space adopted for problem (1) (we take H(Ω) = H01(Ω) for a dirichlet condition on the boundary and H(Ω) = H1(Ω) for a Neuman condition) and u H is the norm on the Sobolev space H ( u H = ∇u in the case of dirichlet condition and u H = u + ∇u in the case of neumann condition)
As in [2] where, we proposed a modified fixed point to a just a semi linear equation, and in [1] where, we applied a modified fixed point to a nonlinear equation, we generalize this method to our equation by mean of solving the following iterative equations: cun+1 − div((D(un) + r(un)), ∇un)∇un+1) + = f (x, y) − div(r(un), ∇un)∇un) + K(un) on Ω un+1 = o on ∂Ω
Summary
To have a good accuracy for the solution we treat, we apply an implicit scheme to this equation and use a modified fixed point technique to linearize the problem of transport equation with a generalized nonlinear reaction and diffusion equation We apply this methods in particular to the the dynamical system of a bio-chemical process. As in [2] where, we proposed a modified fixed point to a just a semi linear equation, and in [1] where, we applied a modified fixed point to a nonlinear equation, we generalize this method to our equation by mean of solving the following iterative equations: cun+1 − div((D(un) + r(un)), ∇un)∇un+1) + (un+1) = f (x, y) − div(r(un), ∇un)∇un) + K(un) on Ω un+1 = o on ∂Ω (2.3) the function r and K are selected such a way that the valuational energy of this equation is a K contraction where, K is very small for reasonable choice of bound M (Generally M Must be less than u0 H ) This small K make the convergence of the method fast and reduce the number of successive iterations in time compared to Newton or Fixed method ones. The convergence to the solution is stable because the energy is a contraction (there is one and only one solution for the problem and the error is controlled)
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