Consistency and stability of a Milstein–Galerkin finite element scheme for semilinear SPDE

Barth, A., Lang, A.: Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises. Appl. Math. Opt. 66(3), 387–413 (2012)

Barth, A., Lang, A.: Multilevel Monte Carlo method with applications to stochastic partial differential equations. J. Comp. Math. 89(18), 2479–2498 (2012)

Barth, A., Lang, A.: $$L^p$$ L p and almost sure convergence of a Milstein scheme for stochastic partial differential equations. Stoch. Process. Appl. 123(5), 1563–1587 (2013)

Barth, A., Lang, A., Schwab, Ch.: Multilevel Monte Carlo method for parabolic stochastic partial differential equations. BIT Numer. Math. 53(1), 3–27 (2013)

Beyn, W.-J., Kruse, R.: Two-sided error estimates for the stochastic theta method. Discrete Contin. Dyn. Syst. Ser. B 14(2), 389–407 (2010)

Burkholder, D.L.: Explorations in martingale theory and its applications. In École d’Été de Probabilités de Saint-Flour XIX–1989. Lecture Notes in Math., Vol. 1464, pp. 1–66. Springer, Berlin 1991.

Chrysafinos, K., Hou, L.S.: Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal. 40(1), 282–306 (2002)

Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)

Elliott, C.M., Larsson, S.: Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Math. Comp. 58(19), 603–630 (1992)

Giles, M.B.: Improved Multilevel Monte Carlo Convergence using the Milstein Scheme. In Monte Carlo and quasi-Monte Carlo methods 2006. Springer, Berlin (2008)

Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

Giles, M.B., Szpruch, L.: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24(4), 1585–1620 (2014)

Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics. Nonstiff problems, 2nd edn. Springer-Verlag, Berlin (1993)

Jentzen, A., Röckner, M.: A Milstein scheme for SPDEs (2012, preprint). arXiv:1001.2751v4

Jentzen, A., Röckner, M.: Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. J. Differen. Equ. 252(1), 114–136 (2012)

Kebaier, A.: Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15(4), 2681–2705 (2005)

Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations Applications of Mathematics (New York). Springer-Verlag, Berlin (1992)

Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 3rd edn. Springer, Berlin (1999)

Kruse, R.: Characterization of bistability for stochastic multistep methods. BIT Numer. Math. 52(1), 109–140 (2012)

Kruse, R.: Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34(1), 217–251 (2014)

Kruse, R.: Strong and Weak Approximation of Stochastic Evolution Equations. Lecture Notes in Mathematics. Springer, Berlin (2014)

Kruse, R., Larsson, S.: Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise. Electron. J. Probab. 17(65), 1–19 (2012)

Lang, A., Chow, P.-L., Potthoff, J.: Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type. Stochastics 82(3), 315–326 (2010)

Lang, A., Chow, P.-L., Potthoff, J.: Erratum: Almost sure convergence of a semi-discrete Milstein scheme for SPDEs of Zakai type. Stochastics 84(4), 561 (2012)

Larsson, S.: Semilinear parabolic partial differential equations: theory, approximation, and application. In New Trends in the Mathematical and Computer Sciences, pp. 153–194. Int. Cent. Math. Comp. Sci. (ICMCS), Lagos 2006.

Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1995). Translated and revised from the 1988 Russian original

Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences. Springer, New York (1983)

Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)

Spijker, M.N.: Stability and convergence of finite-difference methods. Doctoral dissertation vol. 1968. University of Leiden. Rijksuniversiteit te Leiden, Leiden 1968.

Spijker, M.N.: On the structure of error estimates for finite-difference methods. Numer. Math. 18, 73–100 (1971/1972)

Stummel, F.: Approximation Methods in Analysis: Lectures delivered during the Spring Semester. Lecture Notes Series No 35. Matematisk Institut, Aarhus Universitet, Aarhus (1973).

Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, 2nd edn. Springer-Verlag, Berlin (2006)

Wang, X.: An exponential integrator scheme for time discretization of nonlinear stochastic wave equation (2013, preprint). arXiv:1312.5185