Error estimates and free-boundary convergence for a finite difference discretization of a parabolic variational inequality

— We study a free-boundary problem ofbiological interest, for the heat équation ; the problem is reduced to a variational inequality, which is solved by means ofafinite différence method. We aiso give an error estimaté, in the L^-norm, for the solution of the inequality, and a convergence theorem for the discrete free boundaries, The latter follows from a gênerai resuit concerning the behaviour of the free boundaries of "perturbed" problems. (see Th. 1.1). 1. TNTRODTJCTTÖN̂ ~ ~ " " " ~~ ~ ~ " ~ A diffusion-absorption problem (namely, of oxygen in a tissue; see e. g. the book [6], in particular page 319) leads to consider the following freeboundary problem: PROBLEM 1.1: Find two "smooth" functions x, u, such that: x : [0, 1 ] -> R is strictly decreasing, with x(l) = 0, (1.1) and, setting O = {(x,0 | 0 < x < l , 0 O < x ( x ) } , (1.2) it is [Q denoting the closure of Q] : u : Q -* R; u is strictly positive in Q; (1-3) MXJB-II, = 1 in Q; (1.4) u(x, T(X)) = iix(x, T(X)) = 0(0 ̂ x S 1); ux(0? 0 0(0 < t < x(0)); (1.5) t/(x,0) = i ( l x ) 2 J ( O ^ x ^ l ) . (1.6) Because of their relevance, problems like the previous one have been widely studied both from the theoretical and from the numerical point of view; (*) Manuscrit reçu Ie 23 mars 1977. () Laboratório di Analisi Numerica del C.N.R., Pavia. () Ancona University; L.A.N, and G.N.A.F.A. of C.N.R. R.A.Ï.R.O. Analyse numérique/Numerical Analysis, vol. 11, n° 4, 1977 3 1 6 C. BAIOCCHI, G. A. POZZI see e. g. [1, 2, 3, 7, 8, 9, 13]; see also [11, 17, 18, 19] for gênerai results about parabolic free-boundary problems and for further références. As pointed out e. g. in [1, 2, 3, 9], Problem 1.1 can be solved by means of a suitable variational inequality: let R* be the open half-space: R+ = { ( x , O I * e R , * > 0 } ; if we extend u to R^ on setting: t 7 ( X i 0 = j « ( | 4 0 for(|*|,'0eâ (1.7) ( 0 elsewhere, then it is: 17*0; E7xx-C7 r£l; U(UXXUt-1) = 0 inR 2 +; (1.8) l/(x,0) = l/o(x) in R (1.9) [in the particular case of Probl. 1.1, it is V0(x) = -[(l-\x\)]\ (1.10) where 0 = 1/2 (9 + | 0 | ); but we will treat more gênerai initial data]. As we will see, if the initial value Uo is such that UoeHiÇBL); U0(x)^0, VxeR (1.11) [where H (R) is.the usual Sobolev space {veL (R)| v'eL (R) }, and the index c means "with compact support"], then problem (1.8), (1.9) is well posed if we look for a solution U with UeHl'^) (1.12) [say, Us H (R+) ( ) and supp (Ü) is compact in R^]. Moreover, if [as in (1.10)] Uo (x) = Uo (-x)9 and C/g (je) ^ 1, U'o (x) ^ 0 for x > 0, the solution U satisfies U ( x , t) = U(x9 t) and Ut (x, t) ^ 0 , Ux(x9t) £0 in the fîrst quadrant Q = { (x, t) \ x > 0, t > 0 }, so that, setting: Q={(x9t)eQ | U(x,t)>0}; _ w = l / | 5 ; ) x () {i | (x, Oefl}, | we get the solution of Problem 1.1. This theoretical approach was developed in [1] by means of a semi-discretization of a problem like (1.8), (1.9) (actually, in the first quadrant 0 ; in () We follow the notations of [15, 14], to which books we refer also for the properties of Sobolev spaces; in particular, U e H l (R^) means that U, Ux, UXXJ Ut e L 2 (R%). R.A.I.R.O. Analyse numérique/Numerical Analysis DISCRETIZATION OF A PARABOLIC VARIATIONAL INEQUALITY 317 the present paper we will study a complete (i. e., both in x and t ) discretization of (1.8), (1.9), and give convergence theorems both for the family { Uhtk } approximating U and for the famiiy {£lhtk } approximating Q. The approximate solution Uhtk willbe constructed by combining with spline functions the values of discrete solutions (discretization in finite différences, h and k being the mesh-sizes in x and t respectively); the main tooi for the estimâtes will be the maximum principle. The order of convergence will depend, in gênerai, on the smoothness of the initial datum Uo (x); e. g., for a "gênerai" Uo [i. e. satisfying just (1.11)] we will get + , V e > 0 (1.14) [the condition k ~ h being however unnecessary for convergence]. Let us point out the interest of estimâtes like (1.14) in solving free boundary problems. In fact, consider, in a domain Z), a séquence of functions { ƒ„ }n e N converging, in some topology, to a function ƒ In gênerai, no matter of the topology in which ƒ„ —•ƒ (), we cannot be sure that the set of positivity of/„ wilLconvergeLto the .se t of .pasitivitjLof ƒ However, if we know an upper bound for || ƒ„ —ƒ ||L«(D), then we can construct an approximation of the set of positivity of/ In fact, let us prove the following theorem: THEOREM 1.1: Let f, fn (n ~ 1, 2, . . . ) be continuous functions in a domain D, such that: \\fn-f\\L«(D)~Zn^0 (* + GO). (1.15) Let 8„ be such that: 5B > 0 V n ; S„ -> 0 and ^ -> 0 (n -> + oo), (1.16) and let us set: £l = {PeD | / ( P ) > 0 } ; n„ = {PeZ) | ƒ„(?)> Ô.}. (1.17) Then { Qn }„^ly2,... converges to Q, say: Q = lim Qn [in set theoretical sense], (1.18) «-•00 and the convergence is "from the interior", in the sense that there exists n (which can be actually computed) such that Qn^Q, V n ^ n . (1.19) Proof: Let n be such that 8rt ^ zn for n ^ n; when P e Q„, n ^ n, we have f(P) >fm(F)-\\fn-f\\L„ (D) =/„(P)-E„ > on-6„ ^ 0; () e. g., for fn s l/«, ƒ = 0, the set of positivity of f„ is the whole of D for each «, while the set of positivity of ƒ is empty.