Multiscale techniques for parabolic equations

We use the local orthogonal decomposition technique introduced in MA<yen>lqvist and Peterseim (Math Comput 83(290):2583-2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the -norm. We present numerical examples, which confirm our theoretical findings.