In this talk we present recent numerical multiscale methods for linear and nonlinear parabolic problems. The spatial discretization is performed with the finite element heterogenenous multiscale method (FE-HMM). For linear problems we present fully discrete spacetime error estimates for several classes of time integration methods including explicit stabilized integrators [1,2,5]. For nonlinear monotone parabolic problems, in a general $L^p(W^{1,p})$ setting, we prove the convergence of a method that combines the implicit Euler method in time with the FE-HMM in space [4,5]. The upscaling procedure of the method however relies on nonlinear elliptic micro problems. A new linearized scheme [3] that avoids this computational overhead and only involves linear micro problems will finally be discussed.

References:

[1] A. Abdulle, G. Vilmart, Coupling heterogeneous multiscale fem with Runge-Kutta methods for parabolic homogenization problems: a fully discrete spacetime analysis, Math. Models Methods Appl. Sci., 22, 2012.

[2] A. Abdulle, G. Vilmart, PIROCK: A swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise, J. Comput. Phys., 242, 2013.

[3] A. Abdulle, M. Huber and G. Vilmart, Linearized numerical homogenization methods for nonlinear monotone parabolic multiscale problems, to appear in SIAM MMS 2015.

[4] A. Abdulle and M. Huber, Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems: a fully discrete space-time analysis, preprint.

[5] A. Abdulle, Numerical homogenization methods for parabolic monotone problems, preprint.