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- On the finite element method for a nonlocal degenerate parabolic problemPublication . Almeida, Rui M.P.; Antontsev, Stanislav N.; Duque, José C. M.The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k≥1. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.
- On a nonlocal degenerate parabolic problemPublication . Almeida, Rui M.P.; Antontsev, Stanislav N.; Duque, José C. M.Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
- Discrete solutions for the porous medium equation with absorption and variable exponentsPublication . Almeida, Rui M.P.; Antontsev, Stanislav N.; Duque, José C. M.In this work, we study the convergence of the finite element method when applied to the following parabolic equation: 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.