Numerical Analysis of Parabolic p-Laplacian: Approximation of Trajectories

Abstract

Abstract. The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t ∈ [0, ∞) are established in the L 2 , W 1,p norms for the backward Euler and the Crank-Nicholson schemes with the finite element method (FEM). This result extends the existing uniform convergence results for exponentially contractive semigroups generated by some semilinear systems to nonexponentially contractive semigroups generated by some quasilinear systems. 1. Introduction. The parabolic p-Laplacian problem is a mathematical model possessing some important features shared by many practical problems, such as the non-Newtonian fluid flows (see, e.g., [22,23,26]) and the Smagorinsky type meteorology model (see, e.g., [27]). The dynamics of these problems, considered as dissipative dynamical systems, is important. For a general presentation on the dissipative dynamical systems, see, for instance, [13,33]. For the numerical aspects, see [31,32].For semilinear systems, when solutions contract towards each other exponentially with respect to time, several classical numerical solutions approximate the corresponding true solutions uniformly well for t ∈ [0, ∞). This has been confirmed for ordinary differential equations [30], reaction-diffusion equations [17,24,29], and for the Navier-Stokes equations [16]. For semilinear systems, there are many interesting results concerning the long-time numerical approximations. See, among many others references, [7,8,9,12,14,15,16,21,25,31,32] and the references cited therein. However, to the best of our knowledge, no such result is yet available for a nonsemilinear system. The purpose of this article is to extend the uniform-in-time convergence result to a quasi-linear problem, where the rate of contraction for the solutions is only algebraic. See [10]. This model problem covers a class of problems of monotone type. Notice that in [16], the assumption of exponential contraction is used explicitly for the analysis of the Navier-Stokes equations. Also, assuming the (one-sided) Lipschitz condition may imply exponential contraction.New difficulties come from the strong nonlinearity. Some regularity results for the solution used in the previous analysis of the semilinear problems are not available. Also, the semigroup theory is not as easily applied as before. The p-Laplacian operator is not self-adjoint nor commutable with many other operators. For these and other reasons, it is not surprising that the error estimates obtained for such kind of problems may not be optimal. *

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