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— Error bounds introduced in [7] were givenforfully discretized approximate solutions of parabolic équations by thefinite element method. For time discretization the Astable linear v-step methods ( for v = 1 or 2) were used. In this paper the Aostable linear v-step methodsfor any v are used for time discretization. lt is known that Astable methodsfor v = l, 2 are included in the class of Aostable methods. The considération for the elliptic équations is similar to the parabolic équations. Hence, the error bounds for elliptic équations are formulated in this paper too. Résumé. — On a donné en [7] des majorations de l'erreur pour des approximations complètement discrètes d'équations paraboliques par la méthode des éléments finis. On utilisait des méthodes linéaires Astables à v pas (v = 1 ou 2) pour la discrétisation en temps. Dans cet article, on utilise des méthodes linéaires A0-stables à v pas, v quelconque, pour la discrétisation en temps. On sait que les méthodes Astables pour v = l, 2 sont incluses dans la classe des méthodes A0-stables. Les développements étant semblables dans les cas elliptiques et paraboliques, on énonce également dans cet article les majorations d'erreurs pour les équations elliptiques. 1. CONSTRUCTION OF THE FINITE ELEMENT SPACE. NOTATION We consider the /c-regular family {K} h of simplidal isoparametric fmite éléments K introduced by Ciarlet and Raviart [3]. Hence, the simplicial element K e {K } h is the image of the unit rc-simplex Ê. ( K is the closed convex huil of a set î = \J { ât} ) through the unique mapping FK : K -> R n (the mapping FK is i=i ) supposed to be a C*-diffeomorphism) such that FKeP , FK{âi)-=ai k + 1 is a finite dimensional space of functions defined on K with (*) Reçu septembre 1978. f) Technical University, Laborator pocitacich strojû V.U.T., Brno, Tchécoslovaquie. R.A.I.R.O. Analyse numérique/Numerical Analysis, 0399-0516/1979/257/$ 4.00 © Bordas-Dunod 258 J. NEDOMA dim P = N such that Ê is P-unisolvent and JP=>P(1), where for any integer r^O, P(r) is the space of restrictions to K of all polynomials of degree ^ r in n variables x1( . . . , xn) and there exist constants cit O^f ̂ k + 1 , independent of h such that for all h: and sup max|/>,Mx)| ^c4ft'f l ^ i ^ k + 1 (1.1) xetf |a | i 0< — h^\jK{x)\^coh\ (1.2) where a = (a1, . . . , a„), ja | = a t + . . . +a„ and JK(x) is the Jacobian of the mapping FK at the point xeK. To every element K there is associated the finite dimensional space PK (with dim PK = N) of functions ^ = { ^ : ^ ^ ^ ; ^ = P * ( ^ K 1 ) > V P * G P } . (1.3) The X-interpolate rcx w of a given function u : K -> i? is the unique function which satisfies nKuePK, nKu(ai) = u(ai), l^i^N. (1.4) For a /c-regular family { X} h of finite éléments the following interpolation theorem is true (see Ciarlet and Raviart [3], theorem 2, p. 429). LEMMA 1.1 (interpolation theorem): Let ak-reguiarfamily {K}hof simplicial éléments such that P(k)<=P be given. Let fc>^-l. (1.5) Thenfor any integer i such that 0 ̂ i ̂ k +1 , there exists a constant c independent of h such that for any Ke{K}h and for any function ueH (K) we have IU-HKUL^C/I^-'^, .* . (1.6) Here the following notation is used: The norm and the scalar product in the space L{A) is denoted by ||.||0 A and (., . )0 A respectively. H(A)= W^(A)$ m = 0, 1 . . . is a Sobolev space with the norm / = l/2 • where Kt=< I \\*41A)| a | = i R.A.I.R.O. Analyse numérique/Numerical Analysis ELLIPTIC AND PARABOLIC EQUATIONS 259 In the sequel we mean by Q a bounded domain in R with a sufficiently smooth boundary dÇï. Using the way described by Ciarlet and Raviart [3] we define a fc-regular triangulation €h oi£l. Let Qh be the union of a finite number of simplicial éléments K. Every element K = FK(K) is determined by N points altK. We suppose that all points alK belong to Q. The family of éléments constructed in this way is called a triangulation of Q and is denoted by €h. We say that a triangulation êh of Q is /c-regular if: a) the family of all éléments from which the triangulation is formed is fc-regular; b) the geometrical shape of any "face" À of a given element Ke^h must be completely determined by those points ah K which belong to A; c) for the boundary éléments (i. e. for éléments K^Q) of the triangulation êh we have , (1-7) y'eà where c is a constant independent of h and the notation is that of figure.

The finite element solution of elliptic and parabolic equations using simplicial isoparametric elements