Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

Abstract

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u ⁰, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10, 11]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.

Keywords:

• Abstract differential equations
• Analytic C ₀-semigroups
• Banach spaces
• Compact convergence of resolvents
• Condensing operators
• Discretization in space
• Fractional powers of operators
• Hyperbolic equilibrium point
• Parabolic problem
• Semidiscretization
• Theory of shadowing

Mathematics Subject Classification:

• 65J
• 65M
• 65N
• 35J
• 35K