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Convergence Rates for Semi-Discrete Splitting Approximations for Degenerate Parabolic Equations with Source Terms

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Abstract

We study a semi-discrete splitting method for computing approximate viscosity solutions of the initial value problem for a class of nonlinear degenerate parabolic equations with source terms. It is fairly standard to prove that the semi-discrete splitting approximations converge to the desired viscosity solution as the splitting step Δt tends to zero. The purpose of this paper is, however, to consider the more difficult problem of providing a precise estimate of the convergence rate. Using viscosity solution techniques we establish the L∞ convergence rate \(\mathcal{O}(\sqrt{\Delta t})\) for the approximate solutions, and this estimate is robust with respect to the regularity of the solutions. We also provide an extension of this result to weakly coupled systems of equations, and in the case of more regular solutions we recover the “classical” rate \(\mathcal{O}(\Delta t)\) . Finally, we analyze in an example a fully discrete splitting method.

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References

  1. M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner, and P. E. Souganidis, Viscosity Solutions and Applications, Springer-Verlag, Berlin, 1997. Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12–20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C.I.M.E. [C.I.M.E. Foundation].

    Google Scholar 

  2. G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, in Numerical Methods in Finance, Cambridge Univ. Press, Cambridge, 1997, pp. 1–21.

    Google Scholar 

  3. G. Barles, C. Daher, and M. Romano, Convergence of numerical schemes for parabolic equations arising in finance theory, Math. Models Methods Appl. Sci., 5(1) (1995), pp. 125–143.

    Article  Google Scholar 

  4. G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations, M2AN Math. Model. Numer. Anal., 36(1) (2002), pp. 33–54.

    Article  Google Scholar 

  5. G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton–Jacobi–Bellman equations, To appear in SIAM J. Numer. Anal.

  6. G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton–Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21(1) (1990), pp. 21–44.

    Article  Google Scholar 

  7. G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second-order equations, Asymptotic Anal., 4(3) (1991), pp. 271–283.

    Google Scholar 

  8. J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B, 265 (1967), pp. 333–336.

    Google Scholar 

  9. L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications 43, Amer. Math. Soc., Providence, RI, 1995.

    Google Scholar 

  10. F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29(1) (1995), pp. 97–122.

    Google Scholar 

  11. B. Cockburn, G. Gripenberg, and S.-O. Londen, Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations, J. Differential Equations, 170(1) (2001), pp. 180–187.

    Article  Google Scholar 

  12. M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations, 3(6) (1990), pp. 1001–1014.

    Google Scholar 

  13. M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27(1) (1999), pp. 1–67.

    Google Scholar 

  14. M. H. A. Davis, V. G. Panas, and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control Optim., 31(2) (1993), pp. 470–493.

    Article  Google Scholar 

  15. K. Deckelnick and G. Dziuk, Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs, Interfaces Free Bound., 2(4) (2000), pp. 341–359.

    Google Scholar 

  16. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.

    Google Scholar 

  17. H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Comm. PDE., 16(6&7) (1991), pp. 1095–1128.

    Google Scholar 

  18. E. R. Jakobsen, On error bounds for approximation schemes for non-convex degenerate elliptic equations, BIT, 44(2) (2004), pp. 269–285.

    Article  Google Scholar 

  19. E. R. Jakobsen, On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems, Math. Models Methods Appl. Sci. (M3AS), 13(5) (2003), pp. 613–644.

    Article  Google Scholar 

  20. E. R. Jakobsen, W2,∞ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension, Proc. Amer. Math. Soc., 132(11) (2004), pp. 3203–3213.

    Article  MathSciNet  Google Scholar 

  21. E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations, Electron. J. Differential Equations, pages No. 39, 10 pp. (electronic), 2002.

  22. E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 183(2) (2002), pp. 497–525.

    Article  Google Scholar 

  23. E. R. Jakobsen, K. H. Karlsen, and N. H. Risebro, On the convergence rate of operator splitting for Hamilton–Jacobi equations with source terms, SIAM J. Numer. Anal., 39(2) (2001), pp. 499–518 (electronic).

    Article  Google Scholar 

  24. E. R. Jakobsen, K. H. Karlsen, and N. H. Risebro, On the convergence rate of operator splitting for weakly coupled systems of Hamilton–Jacobi equations, in Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, II (Magdeburg, 2000), Vol. 141 of Internat. Ser. Numer. Math. 140, Birkhäuser, Basel, 2001, pp. 553–562.

    Google Scholar 

  25. K. H. Karlsen and K.-A. Lie, An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. Numer. Anal., 19(4) (1999), pp. 609–635.

    Article  Google Scholar 

  26. N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman’s equations, Algebra i Analiz, 9(3) (1997), pp. 245–256.

    Google Scholar 

  27. N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients, Probab. Theory Related Fields, 117(1) (2000), pp. 1–16.

    Article  Google Scholar 

  28. H. J. Kuo and N. S. Trudinger, Discrete methods for fully nonlinear elliptic equations, SIAM J. Numer. Anal., 29(1) (1992), pp. 123–135.

    Article  Google Scholar 

  29. H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer-Verlag, New York, 1992.

    Google Scholar 

  30. J. O. Langseth, A. Tveito, and R. Winther, On the convergence of operator splitting applied to conservation laws with source terms, SIAM J. Numer. Anal., 33(3) (1996), pp. 843–863.

    Article  Google Scholar 

  31. G. T. Lines, M. L. Buist, P. Grøttum, A. J. Pullan, J. Sundnes, and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology, Comput. Vis. Sci., 5 (2003), pp. 215–239.

    Article  Google Scholar 

  32. G. T. Lines, P. Grøttum, and A. Tveito, Modeling the electrical activity of the heart: A bidomain model of the ventricles embedded in a torso, Comput. Vis. Sci., 5 (2003), pp. 195–213.

    Article  MathSciNet  Google Scholar 

  33. Z. Qu and A. Garfinkel, An advanced algorithm for solving partial differential equation in cardiac conduction, IEEE Transactions of Biomedical Engineering, 46(9) (1999), pp. 1166–1168.

    Article  Google Scholar 

  34. P. E. Souganidis, Existence of viscosity solutions of Hamilton–Jacobi equations, J. Differential Equations, 56(3) (1985), pp. 345–390.

    Article  Google Scholar 

  35. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), pp. 506–517.

    Article  Google Scholar 

  36. J. Sundnes, G. T. Lines, and A. Tveito, A second-order scheme for solving the coupled bidomain and forward problem, Technical Report, SIMULA Research Laboratory, 2002.

  37. T. Tang, Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms, SIAM J. Numer. Anal., 35(5) (1998), pp. 1939–1968 (electronic).

    Article  Google Scholar 

  38. T. Tang and Z. H. Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal., 32(1) (1995), pp. 110–127.

    Article  Google Scholar 

  39. Y. Zhan, Viscosity solution theory of a class of nonlinear degenerate parabolic equations. I. Uniqueness and existence of viscosity solutions, Acta Math. Appl. Sinica (English Ser.), 13(2) (1997), pp. 136–144.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    E. R. Jakobsen

  2. Centre of Mathematics for Applications (CMA), University of Oslo, 0316 Oslo, Norway, 1325, Lysaker, Norway

    K. H. Karlsen

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  1. E. R. Jakobsen
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  2. K. H. Karlsen
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Correspondence to E. R. Jakobsen.

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AMS subject classification (2000)

65M15, 35K65, 35K55, 49L25.

Received July 2004. Revised November 2004. Communicated by Per Lötstedt.

K. H. Karlsen: This work is supported by the Norwegian Research Council via grant no. 121531/410 (ERJ), the BeMatA program (KHK), and an Outstanding Young Investigators Award (KHK), and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282 (KHK).

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Jakobsen, E.R., Karlsen, K.H. Convergence Rates for Semi-Discrete Splitting Approximations for Degenerate Parabolic Equations with Source Terms. Bit Numer Math 45, 37–67 (2005). https://doi.org/10.1007/s10543-005-2641-0

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