Discrete solutions for the porous medium equation with absorption and variable exponents

In this work, we study the convergence of the finite element method when applied to the following parabolic equation: ut=div(|u|γ(x,t)∇u)−λ|u|σ(x,t)−2u+f,x∈Ω⊂Rd,t∈]0,T]. Since the equation may be of degenerate type, we use an approximate problem, regularized by introducing a parameter ε. We prove, u...

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Bibliographic Details
Published in:Mathematics and computers in simulation 2017-07, Vol.137, p.109-129
Main Authors: Almeida, Rui M.P., Antontsev, Stanislav N., Duque, José C.M.
Format: Article
Language:English
Subjects:
Finite element method
Porous medium equation
Variable exponents
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Summary:In this work, we study the convergence of the finite element method when applied to the following parabolic equation: ut=div(|u|γ(x,t)∇u)−λ|u|σ(x,t)−2u+f,x∈Ω⊂Rd,t∈]0,T]. Since the equation may be of degenerate type, we use an approximate problem, regularized by introducing a parameter ε. We prove, under certain conditions on γ, σ and f, that the weak solution of the approximate problem converges to the weak solution of the initial problem, when the parameter ε tends to zero. The convergence of the discrete solutions for the weak solution of the approximate problem is also proved. Finally, we present some numerical results of a MatLab implementation of the method.
ISSN:0378-4754
1872-7166
DOI:10.1016/j.matcom.2016.12.008