On Well-Posedness of Difference Schemes for Abstract Elliptic Problems in L^p([0, T];E) Spaces

Abstract

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 spaces with the factor is obtained. In the case of UMD space E _n , we establish a coercive inequality for the same scheme in under the condition of R-boundedness.

Keywords:

• Abstract differential equations
• Abstract elliptic problem
• Analytic C ₀-semigroups
• Banach spaces
• Coercive inequality
• Difference schemes
• Discrete semigroups
• Maximal regularity
• Semidiscretization
• UMD spaces
• Well-posedness

AMS Subject Classification:

• 65J
• 65N
• 35J
• 47D

ACKNOWLEDGMENTS

Research partially supported by Russian Foundation for Basic Research 04-01-00026, by grant CNPq 307753/2004-2, and by grant SFB 701, “Spectral Structures and Topological Methods in Mathematics,” Bielefeld University.