Variable Time-Step Discretization of Degenerate Evolution Equations in Banach Spaces

• Published in: Numerical functional analysis and optimization Vol. 24; no. 3-4; pp. 391 - 426
• Main Author: Bassetti, Federico
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 08.01.2003
• Subjects: AMS Subject Classification: 35A40; Backward Euler method; Degenerate evolution equations; Differential inclusion; Error estimates
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1081/NFA-120022930

This work is concerned with an abstract differential inclusion of the form where is a linear, continuous, symmetric and monotone operator defined over a separable Banach space V, and ∂φ is the subdifferential of a proper, convex, l.s.c, positive real function. We consider an approximation of the previous equation by a backward Euler method with variable time-step. Under suitable hypothesis of coercivity we prove that the discrete solution converges uniformly to a strong solution of the equation, in the seminorm induced by B, as the maximum of the time steps goes to 0. We derive computable estimates of the discretization error, which are optimal w.r.t. the order and impose no constrains between consecutive time steps. In addition we prove some regularity and uniqueness results for the solution. Finally we extend some of the previous results to the case in which ∂φ is perturbed by a Lipschitz map.