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Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator. (English) Zbl 1172.65026
The paper is concerned with the initial value problem
$u^{\prime}+Au=f \quad \text{in }(0,T), \qquad u(0)=u_0,$
where the nonlinear operator $$A:V \to V^{*}$$ acting on a Gelfand triple $$V\subseteq H \subseteq V^{*}$$ is supposed to be a monotone and coercive potential operator that fulfills a growth condition. On the variable time grid $$\mathbb{I}: 0=t_0 < t_1< \dots < t_N=T$$ $$(N \in \mathbb{N})$$ with $$\tau_n=t_n-t_{n-1}$$ $$(n=1,2,\dots,N)$$, $$r_n=\frac{\tau_{n}}{\tau_{n-1}}$$ $$(n=2,3,\dots,N)$$, $$\tau_{\max}=\max_{n=1,2,\dots,N}\tau_{n}$$, $$r_{\max}=\max(\max_{n=2,3,\dots,N}r_n,1)$$ the two-step temporal semidiscretization for the computation of a time discrete solution $$u^{n}\approx u(t_n)$$ with an initial implicit Euler step
$Du^{n}+Au^{n}=f^{n}, \quad n=1,2,\dots,N$
is considered, where
$Du^{1}=\frac{1}{\tau_1}\left(u^{1}-u^{0}\right),\quad Du^{n}=\frac{1}{\tau_1}\left(\frac{1+2r_n}{1+r_n}u^{n}-(1+r_n)u^{n-1}+ \frac{r_n^2}{1+r_n}u^{n-2}\right).$
It is proven that a piecewise linear prolongation of the discrete solution converges to the weak solution of the initial problem provided that the ratios of adjacent step sizes are close to 1 and do not vary too much.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 34G20 Nonlinear differential equations in abstract spaces 35K90 Abstract parabolic equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
RODAS
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##### References:
  Akrivis, G., Crouzeix, M.: Linearly implicit methods for nonlinear parabolic equations. Math. Comput. 73(246), 613–635 (2004) · Zbl 1045.65079  Akrivis, G., Crouzeix, M., Makridakis, C.: Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comput. 67(222), 457–477 (1998) · Zbl 0896.65066  Akrivis, G., Crouzeix, M., Makridakis, C.: Implicit-explicit multistep methods for quasilinear parabolic equations. Numer. Math. 82(4), 521–541 (1999) · Zbl 0936.65118  Akrivis, G., Makridakis, C., Nochetto, R.H.: A posteriori error estimates for the Crank-Nicolson method for parabolic equations. Math. Comput. 75(254), 511–531 (2006) · Zbl 1101.65094  Axelsson, O., Gololobov, S.V.: Stability and error estimates for the -method for strongly monotone and infinitely stiff evolution equations. Numer. Math. 89(1), 31–48 (2001) · Zbl 0994.65094  Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff Int. Publ., Leyden (1976) · Zbl 0328.47035  Becker, J.: A second order backward difference method with variable steps for a parabolic problem. BIT 38(4), 644–662 (1998) · Zbl 0923.65050  Brayton, R.K., Conley, C.S.: Some results on the stability and instability of the backward differentiation methods with non-uniform time steps. In: Topics in Numerical Analysis, Proc. Roy. Irish Acad. Conf. University Coll., Dublin, 1972, pp. 13–33. Academic Press, London (1973)  Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam (1973) · Zbl 0252.47055  Brézis, H.: Analyse fonctionnelle: Théorie et applications. Dunod, Paris (1999)  Byrne, G.D., Hindmarsh, A.C.: Stiff ODE solvers: a review of current and coming attractions. J. Comput. Phys. 70(1), 1–62 (1987) · Zbl 0614.65078  Calvo, M., Grigorieff, R.D.: Time discretisation of parabolic problems with the variable 3-step BDF. BIT 42(4), 689–701 (2002) · Zbl 1014.65086  Calvo, M.P., Palencia, C.: A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102(3), 367–381 (2006) · Zbl 1087.65054  Calvo, M., Grande, T., Grigorieff, R.D.: On the zero-stability of the variable order variable stepsize BDF-formulas. Numer. Math. 57(1), 39–50 (1990) · Zbl 0696.65077  Calvo, M., Montijano, J.I., Rández, L.: A 0-stability of variable stepsize BDF methods. J. Comput. Appl. Math. 45(1–2), 29–39 (1993) · Zbl 0789.65064  Calvo, M., Montijano, J.I., Rández, L.: On the change of step size in multistep codes. Numer. Algorithms 4(3), 283–304 (1993) · Zbl 0779.65055  Crouzeix, M., Thomée, V.: On the discretization in time for semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49(180), 359–377 (1987) · Zbl 0632.65097  Emmrich, E.: Error of the two-step BDF for the incompressible Navier-Stokes problem. M2AN Math. Model. Numer. Anal. 38(5), 757–764 (2004) · Zbl 1076.76054  Emmrich, E.: Stability and convergence of the two-step BDF for the incompressible Navier-Stokes problem. Int. J. Nonlinear Sci. Numer. Simul. 5(3), 199–210 (2004) · Zbl 1076.76054  Emmrich, E.: Stability and error of the variable two-step BDF for semilinear parabolic problems. J. Appl. Math. Comput. 19(1–2), 33–55 (2005) · Zbl 1082.65086  Emmrich, E.: Convergence of a time discretization for a class of non-Newtonian fluid flow. Commun. Math. Sci. 6(4), 827–843 (2008) · Zbl 1160.76034  Emmrich, E.: Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations. Comput. Methods Appl. Math. (to appear) · Zbl 1169.65046  Emmrich, E.: Variable time-step -scheme for nonlinear evolution equations governed by a monotone operator (submitted) · Zbl 1177.65075  Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin (1974) · Zbl 0289.47029  García-Archilla, B.: Some practical experience with the time integration of dissipative equations. J. Comput. Phys. 122(1), 25–29 (1995) · Zbl 0854.65078  García-Archilla, B., de Frutos, J.: Time integration of the non-linear Galerkin method. IMA J. Numer. Anal. 15(2), 221–244 (1995) · Zbl 0823.65091  Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1981) · Zbl 0441.65081  González, C., Palencia, C.: Stability of Runge-Kutta methods for quasilinear parabolic problems. Math. Comput. 69(230), 609–628 (2000) · Zbl 0941.65098  González, C., Ostermann, A., Palencia, C., Thalhammer, M.: Backward Euler discretization of fully nonlinear parabolic problems. Math. Comput. 71(237), 125–145 (2002) · Zbl 0991.65087  Grigorieff, R.D.: Stability of multistep-methods on variable grids. Numer. Math. 42(3), 359–377 (1983) · Zbl 0554.65051  Grigorieff, R.D.: Time discretization of semigroups by the variable two-step BDF method. In: Numerical Treatment of Differential Equations, Sel. Pap. 5th Int. Semin., NUMDIFF-5, Halle, 1989. Teubner-Texte Math., vol. 121, pp. 204–216 (1991)  Grigorieff, R.D.: On the variable grid two-step BDF method for parabolic equations. Preprint 426, FB Mathem., TU Berlin (1995)  Grigorieff, R.D., Paes-Leme, P.J.: On the zero-stability of the 3-step BDF-formula on nonuniform grids. BIT 24(1), 85–91 (1984) · Zbl 0554.65052  Hairer, E., Wanner, G.: Solving Ordinary Differential Equations, II: Stiff Problems. Springer, Berlin (1996) · Zbl 0859.65067  Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations, I: Nonstiff Problems. Springer, Berlin (1993) · Zbl 0789.65048  Hansen, E.: Runge-Kutta time discretizations of nonlinear dissipative evolution equations. Math. Comput. 75(254), 631–640 (2006) · Zbl 1098.65058  Hansen, E.: Convergence of multistep time discretizations of nonlinear dissipative evolution equations. SIAM J. Numer. Anal. 44(1), 55–65 (2006) · Zbl 1118.65055  Hansen, E.: Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations. J. Comput. Appl. Math. 205(2), 882–890 (2007) · Zbl 1220.65123  Hass, R., Kreth, H.: Stabilität und Konvergenz von Mehrschrittverfahren zur numerischen Lösung quasilinearer Anfangswertprobleme. Z. Angew. Math. Mech. 54, 353–358 (1974) · Zbl 0296.65040  Hill, A.T., Süli, E.: Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations. Math. Comput. 64(211), 1097–1122 (1995) · Zbl 0830.65049  Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20(4), 633–667 (2000) · Zbl 0982.76022  Kreth, H.: Time-discretisations for nonlinear evolution equations. In: Numerical Treatment of Differential Equations in Applications, Proc. Meeting Math. Res. Center, Oberwolfach, 1977. Lect. Notes Math., vol. 679, pp. 57–63. Springer, Berlin (1978)  Kreth, H.: Ein Zwei-Schritt-Differenzenverfahren zur Berechnung strömungsabhängiger Ausbreitungsvorgänge. Z. Angew. Math. Phys. 29, 12–22 (1978) · Zbl 0395.65030  Le Roux, M.-N.: Méthodes multipas pour des équations paraboliques non linéaires. Numer. Math. 35, 143–162 (1980) · Zbl 0463.65067  Le Roux, M.-N.: Variable step size multistep methods for parabolic problems. SIAM J. Numer. Anal. 19(4), 725–741 (1982) · Zbl 0483.65033  Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969)  Lubich, C., Ostermann, A.: Runge-Kutta approximation of quasi-linear parabolic equations. Math. Comput. 64(210), 601–627 (1995) · Zbl 0832.65104  Lubich, C., Ostermann, A.: Linearly implicit time discretization of non-linear parabolic equations. IMA J. Numer. Anal. 15(4), 555–583 (1995) · Zbl 0834.65092  Lubich, C., Ostermann, A.: Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: Nonsmooth-data error estimates and applications to long-time behaviour. Appl. Numer. Math. 22(1–3), 279–292 (1996) · Zbl 0872.65090  Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995) · Zbl 0816.35001  Makridakis, C., Nochetto, R.H.: A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104(4), 489–514 (2006) · Zbl 1104.65091  Moore, P.K.: A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31(1), 149–169 (1994) · Zbl 0798.65089  Nochetto, R.H., Savaré, G.: Nonlinear evolution governed by accretive operators in Banach spaces: error control and applications. Math. Models Methods Appl. Sci. 16(3), 439–477 (2006) · Zbl 1098.65092  Nochetto, R.H., Savaré, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53(5), 525–589 (2000) · Zbl 1021.65047  Ostermann, A., Thalhammer, M.: Convergence of Runge-Kutta methods for nonlinear parabolic equations. Appl. Numer. Math. 42(1–3), 367–380 (2002) · Zbl 1004.65093  Ostermann, A., Thalhammer, M., Kirlinger, G.: Stability of linear multistep methods and applications to nonlinear parabolic problems. Appl. Numer. Math. 48(3–4), 389–407 (2004) · Zbl 1041.65073  Palencia, C., García-Archilla, B.: Stability of linear multistep methods for sectorial operators in Banach spaces. Appl. Numer. Math. 12(6), 503–520 (1993) · Zbl 0784.65055  Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005) · Zbl 1087.35002  Rulla, J.: Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33(1), 68–87 (1996) · Zbl 0855.65102  Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002) · Zbl 1254.37002  Slodička, M.: Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data. Comment. Math. Univ. Carol. 32(4), 703–713 (1991) · Zbl 0755.65095  Slodička, M.: Semigroup formulation of Rothe’s method: application to parabolic problems. Comment. Math. Univ. Carol. 33(2), 245–260 (1992) · Zbl 0756.65121  Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996) · Zbl 0869.65043  Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997) · Zbl 0871.35001  Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006) · Zbl 1105.65102  Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/A: Linear Monotone Operators, II/B: Nonlinear Monotone Operators. Springer, New York (1990) · Zbl 0684.47029  Zlámal, M.: Finite element methods for nonlinear parabolic equations. R.A.I.R.O. Anal. Numér. 11(1), 93–107 (1977) · Zbl 0385.65049
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