Multiscale methods for linear and nonlinear parabolic problems

Assyr Abdulle
Date and time
Workshop - Multiscale numerical methods for differential equations

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.


[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.