Homogenization of parabolic equations with a continuum of space and time scales. (English) Zbl 1170.34037

Owhadi, Houman; Zhang, Lei

SIAM J. Numer. Anal. 46, No. 1, 1-36 (2008).

The authors consider a linear parabolic equation in divergence form defined via a uniformly elliptic and symmetric matrix \(a\), with the aim to give some numerical approximation of the solution. In spite of the title the paper is not a paper about some result in homogenization, but deals with numerical approximation of a fixed operator which is defined via a matrix whose coefficients \(a_{ij}\) belong to \(L^{\infty}(\Omega)\), \(\Omega \subset {\mathbb R}^n\) (possibly oscillating and not regular).
The natural space in which a solution of the equation \(u_t-\text{div}(a\cdot\nabla u)=g\), \(g\in L^2(\Omega\times (0,T))\), with \(u=0\) in \(\partial\Omega\times (0,T)\) lives is \(\{u\in L^2(0,T; H^1_0(\Omega))\mid u_t\in L^2(0,T; H^{-1}(\Omega))\}\).
The authors consider a change of variables, called caloric coordinates, defined as the solutions
\[ \begin{alignedat}{2}{\partial_t F_i} - \text{div}(a\cdot\nabla F_i)&= 0 &\quad &\text{in }\Omega \times(0,T),\\ F_i&= x_i &\quad &\text{on }\partial\Omega\times(0,T),\\ \text{div}(a\cdot \nabla F_i)&= 0 &\quad &\text{in }\Omega\times\{0\},\end{alignedat} \] \(i=1,\dots n\). The authors prove that, if \(\Omega\) is convex and under some assumption on the trace of \(\langle a \cdot \nabla F, \nabla F \rangle\), the function \(\tilde u := u \circ F^{-1}\) belong to the space \({\mathcal Z}= \{v\in L^2(0,T; H^2(\Omega))\mid v_t\in L^2(0,T; L^2(\Omega)) \}\) and give some estimate in this space depending (only) on the data. Defining finite dimensional spaces of functions of the type \(\varphi \circ F\) the authors prove several results of convergence of the approximated solutions to the solution of the original problem.

Reviewer: Fabio Paronetto (Padova)

MSC:

34D10	Perturbations of ordinary differential equations
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
68P30	Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
60F05	Central limit and other weak theorems
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations
35A35	Theoretical approximation in context of PDEs

Keywords:

multiscale problem; compensation; upscaling; compression; equation in divergence form; numerical approximation; caloric coordinates