# Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations

## Abstract

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the *L*^{2} space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the *N* best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best *N* term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.

## Keywords

Stochastic parabolic two-scale pdes Generalized polynomial chaos approximation Regularity Analyticity Best*N*-term approximation

## Mathematics Subject Classification (2010)

35R60 35K20 35B27 35A35## Notes

### Funding Information

The research is supported by the Singapore MOE AcRF Tier 1 grant RG30/16 and the MOE Tier 2 grant MOE2017-T2-2-144.

## References

- 1.Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal.
**23**(6), 1482–1518 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bakhvalov, N., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media. Mathematics and its Applications (Soviet Series), vol. 36. Kluwer Academic Publishers Group, Dordrecht (1989). Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. LeĭtesCrossRefGoogle Scholar
- 3.Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures Studies in Mathematics and its Applications, vol. 5. North-Holland Publishing Co., Amsterdam (1978)zbMATHGoogle Scholar
- 4.Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal.
**40**(4), 1585–1620 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Cohen, A., DeVore, R.: Approximation of high-dimensional parametric PDEs. Acta Numer.
**24**(1), 1–159 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Cohen, A., DeVore, R., Schwab, Ch.: Convergence rates of best
*N*-term Galerkin approximations for a class of elliptic sPDEs. Found. Comp. Math.**10**(6), 615–646 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Cohen, A., DeVore, R., Schwab, Ch.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl.
**9**, 11–47 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Grisvard, P.: Elliptic Problems in Nonsmooth Domains Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
- 9.Hoang, V.H., Schwab, Ch.: High-dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul.
**3**(1), 168–194 (2004/05)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hoang, V.H., Schwab, Ch.: Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs. Anal. Appl.
**11**, 1350001 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Hoang, V.H., Schwab, Ch.: Sparse tensor Galerkin discretization of parametric and random parabolic PDEs – analytic regularity and generalized polynomial chaos approximation. SIAM J. Math. Anal.
**45**, 3050–3083 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]CrossRefGoogle Scholar
- 13.Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces Graduate Studies in Mathematics, vol. 96. American Mathematical Society, Providence, RI (2008)CrossRefGoogle Scholar
- 14.Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.
**20**(3), 608–623 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Schwab, Ch., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer.
**20**, 291–467 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Tan, W.C., Hoang, V.H.: High dimensional finite element method for multiscale nonlinear monotone parabolic equations. J. Comput. Appl. Math.
**345**, 471–500 (2019)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Tan, W.C., Hoang, V.H.: High dimensional finite elements for time-space multiscale parabolic equations. Adv. Comput. Math.
**45**, 1291–1327 (2019)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). Translated from the German by C. B. Thomas and M. J. ThomasCrossRefzbMATHGoogle Scholar
- 19.Xia, B.X., Hoang, V.H.: High dimensional finite elements for multiscale wave equations. Multiscale Model Simul.
**12**(4), 1622–1666 (2014)MathSciNetCrossRefzbMATHGoogle Scholar