Erich Carelli’s research while affiliated with University of Tübingen and other places

We study finite-element-based space-time discretizations of the incompressible Navier–Stokes equations with noise. In three dimensions, sequences of numerical solutions construct weak martingale solutions for vanishing discretization parameters. In the two-dimensional case, numerical solutions converge to the unique strong solution.

We consider the numerical approximation of mild solutions of the stochastic, Hilbert space-valued heat equation with a uniformly elliptic operator A:D A → L 2𝒟 and a symmetric operator Q:L 2𝒟→ L 2𝒟 with finite trace. We apply different domain decomposition algorithms based on explicit and implicit time-stepping, together with a finite element and backward Euler discretization to solve the problem, and derive optimal strong and weak rates of convergence. For this purpose, and due to the interplay of limited regularity in time of the driving noise and the splitting character of the scheme, a well-known deterministic domain decomposition algorithm requires modifications to prove an optimal weak rate of convergence.

We show strong convergence with rates for an implicit time discretization, a semiimplicit time discretization, and a related finite element based space-time discretization of the incompressible Navier-Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset Ω κ of the sample space Ω, where corresponding paths are bounded in a proper function space, and ℙ[Ω κ ] → 1 holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.

For the stochastic incompressible time-dependent Stokes equation, we study different time-splitting methods that decouple the computation of velocity and pressure iterates in every iteration step. Optimal strong convergence is shown for Chorin's time-splitting scheme in the case of solenoidal noise, while computational counterexamples show poor convergence behavior in the case of general stochastic forcing. This suboptimal performance may be traced back to the nonregular pressure process in the case of general noise. A modified version of the deterministic time-splitting method that distinguishes between the deterministic and stochastic pressure removes this deficiency, leading to optimal convergence behavior.

We study equations to describe incompressible generalized Newtonian fluids, where the extra stress tensor satisfies a nonstandard anisotropic asymptotic growth condition. An implicit finite element discretization and a simple, fully practical fixed-point scheme with proper thresholding criterion are proposed, and convergence toward weak solutions of the limiting problem is shown. Computational experiments are included, which motivate nontrivial fluid flow behavior.

Projection methods are efficient operator-splitting schemes to approximate solutions of the incompressible Navier-Stokes equations. As a major drawback, they introduce spurious layers, both in space and time. In this paper, we survey convergence results for higher order projection methods, in the presence of only strong solutions of the limiting problem; in particular, we highlight concomitant difficulties in the construction process of accurate higher order schemes, such as limited regularities of the limiting solution, and a lack of accurate initial data for the pressure. Computational experiments are included to compare the presented schemes, and illustrate the difficulties mentioned.