On the convergence of finite difference schemes for the heat equation with concentrated capacity

Abstract

We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is c (x) = 1 + Kδ (x − ξ). K = const > 0 represents the magnitude of the heat capacity concentrated at the point x = ξ. An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Subject Classification (1991): 65M06, 65M12, 65M15 Mathematics

Finite difference approximation of strong solutions of a parabolic interface problem on disconnected domains

“…Section 2 is devoted to the analysis of the existence and the uniqueness of the strong solution of IBVP (1)- (6). In Section 3 we introduce a FDS approximating IBVP (1)-(6) and investigate its convergence.…”

“…The numerical methods designed for smooth solutions do not work efficiently for interface problems. Problems of this type we considered in [6,7] There exists another similar type of problems whose solutions are defined in two (or more) disconnected domains. For example, such situation occurs when the solution in the intermediate region is known or can be determined from a simpler equation.…”

Finite Difference Approximation of an Elliptic Interface Problem with Variable Coefficients

“…In [2] a review of results on numerical solution of two-dimensional elliptic and parabolic interface problems in recent years is presented. In [10], [11], [12], [19] convergence of finite difference method for different elliptic, parabolic and hyperbolic problems in which the solution is continuous on the interface curves, while the flow is discontinuous, is studied.…”

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

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