A finite difference solution to a two-dimensional parabolic inverse problem

Abstract

In this paper, we consider the problem of finding u = u( x, y, t) and p = p( t) which satisfy u t = u xx + u yy + p( t) u + ϕ in R × [0, T], u( x, y, 0) = f( x, y), ( x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u( x ∗, y ∗, t) = E( t), 0 < t ⩽ T, where E( t) is known and ( x ∗, y ∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L ∞ norm. The convergence orders of u and p are of O( τ + h 2). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.