We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show...

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.

An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega$ (E) ${\left(t{u}_t\right)}_t={\sum }_{i,j=1}{\left(a_{ij}\left(x\right)u_{x_i}\right)}_{x_j}-a_0\left(x\right)u+f\left(u\right)$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega$ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\to 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^{2,1}$ is shown in the linear case without assuming $u_{tt}$ (which can blow up as $t\to 0$ is...