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

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy ut = uxx + uyy + 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 + h2). 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.