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Numerical solutions of some parabolic inverse problems. (English) Zbl 0709.65105

Motivated by a class of parabolic inverse problems in which an unknown function p(t) and the solution u(x,t) itself are to be determined the authors study the parabolic equation (with \(x_ 0\) a fixed point in [0,1]) \(u_ t=a(x,t,u,u_ x,u(x_ 0,t))u_{xx}+b(x,t,u,u_ x,u_ x(x_ 0,t))\) for \(0<x<1\), \(0<t\leq T\), with given initial and boundary values of u. The functions a(x,t,u,p,q) and b(x,t,u,p,q) are assumed to be \(C^ 2\) functions, further it is assumed that \(a\geq a_ 0>0\), and \(| \nabla_{u,p,q}a| +\nabla u_{u,p,q}b| \leq A_ 0.\)
The authors give a variational form (with zero initial and boundary values of u) for \(\nu =u_ x\) as unknown function. They describe in detail a continuous Galerkin approximation (discretization in x only) and the application of the Crank-Nicolson method for approximation of the solution of the resulting system of ordinary differential equations. For both discretizations they derive error estimates.
Reviewer: R.Gorenflo

MSC:

65Z05 Applications to the sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35R30 Inverse problems for PDEs
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