Solutions of variational PDEs can often be interpreted as local minima of an associated energy functional (local minima being special instances of critical points) arising in applications such as physics and mechanics. Classical approaches typically focus solely on discretizations of the underlying PDEs. By contrast, we present a perspective that explicitly incorporates the underlying energy structure into the numerical approximation, thereby naturally exploiting more of the available structure of the problem. Specifically, we study linearized iterative Galerkin schemes that guarantee energy reduction at every iteration and give rise to an energy-based paradigm for adaptive finite element discretizations, called variational adaptivity. Applications to semilinear elliptic PDEs from ecology and quantum chemistry illustrate the proposed approach.

We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm approximates the solution of the SDE in the $L^2$-sense without the curse of dimensionality. The latter means that its computational cost grows at most polynomially in both the dimension and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate its applicability by approximating McKean--Vlasov SDEs in dimensions up to 1000.

We study the regularity of discrete solutions to the Poisson problem. This is related to the properties of the Ritz projection. We present two different approaches via Greens functions and Caccioppoli estimates.