In this thesis we look at numerical methods for approximating linear partial differential equations. The thesis is structured into two parts. In the first part, we study the construction of preconditioners for discretized operators using the concept of operator preconditioning. The idea is to precondition the discretized operator by a discretized operator of opposite order. It turns out that in order to get a uniformly well-conditioned system, as well as a preconditioner that can be implemented efficiently, the second discretization has to be carefully chosen dependent on the first one. We apply this idea to construct optimal preconditioners for problems of negative and positive order. For problems of negative order, we additionally introduce a multi-level type operator that both fulfills the role of the opposite order operator and can be applied in optimal linear complexity. In the second part of the thesis, we discuss the (adaptive) numerical solution of parabolic evolution equations, e.g. the heat equation, written in a simultaneous space-time variational formulation. We propose an r-linearly converging adaptive method that is able to resolve singularities locally in space and time. We achieve this by using trial- and test spaces that are given as the spans of wavelets-in-time tensorized with (locally refined) finite element spaces-in-space. We also discuss a linear complexity implementation of this method. Lastly, we introduce an adaptive space-time boundary element method for the solution of the homogeneous heat equation with prescribed initial condition and Dirichlet data.