We consider semilinear problems of the form u = Au + f (u), where A generates an exponentially decaying compact analytic C 0 -semigroup in a Banach space E , f : E → E is differentiable globally Lipschitz and bounded (E = D((−A) ) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.Keywords Abstract parabolic equations; Compact convergence of resolvents; General approximation scheme; Upper and lower semicontinuity of attractors.

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This paper is devoted to the numerical analysis of abstract elliptic differential equations in L p ([0, T ]; E ) spaces. The presentation uses general approximation scheme and is based on C 0 -semigroup theory and a functional analysis approach. For the solutions of difference scheme of the second-order accuracy, the almost coercive inequality in L p n ([0, T ]; E n ) spaces with the factor min |ln 1 n |, 1 + |ln B n B(En ) | is obtained. In the case of UMD space E n , we establish a coercive inequality for the same scheme in L p n ([0, T ]; E n ) under the condition of R -boundedness.

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