The Finite Element Method in a Family of Improperly Posed Problems

The numerical solution of the Cauchy problem for elliptic equations is considered. We reformulate the original problem as a variational inequality problem, which we solve using the finite element method. Moreover, we prove the convergence of the approximate solution. Let <$> be a bounded open set in the space R" and d6^ be the boundary of fy. Then ßr = ty X (0, T) is a bounded open set in Rn+I. We discuss the following boundary value problem for the elliptic equation: (I) U\d<¡¡)XlO,T] = 0, "Uo = /(*)> du dt Here A,,, X are functions of x and, moreover, = g(x). 0 Ktf + *l + • • • +tf) < 2 A,fâ < vx(t2 + i2 + ■ ■ ■ H2), vx > v > 0; A > Ag > 0. v, vx, Xq are constants. If there are no added restrictions to the solutions of (I), J. Hadamard [1] has pointed out that the solution of (I) is not continuously dependent on the Cauchy data. So problem (I) is an improperly posed problem. As the famous example of J. Hadamard has shown, it is impossible to solve this improperly posed problem by the classical theory of partial differential equations. But these types of problems arise naturally in many kinds of practical problems and therefore have required the attention of many mathematicians. First, M. M. Lavrentiev [2] has discussed bounded solutions of the Laplace equation in a special two-dimensional domain. These solutions are dependent on the Cauchy data continuously. After this L. E. Payne [3], [4] studied solutions of more general second-order elliptic equations, which are dependent on the Cauchy data continuously. Of course, it is necessary to add some restrictions to the domains and the solutions. In 1975 L. E. Payne outlined this problem in [5]. Received December 5, 1980; revised April 27, 1981. 1980 Mathematics Subject Classification. Primary 65M30, 35R25. © 1982 American Mathematical Society 0025-5718/82/0000-0469/$03.75 55 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use