An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions

A new adaptive multilevel approach, for linear parabolic partial dif-ferential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A the-ory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly per-turbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of precondition-ing. A multilevel nodal basis preconditioner is derived, which allow