Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement

Recurrent Neural Networks as Optimal Mesh Refinement Strategies

We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input size, i.e., number of elements of the mesh. Moreover, for a general class of PDEs with solutions which are well-approximated by deep neural networks, we show that an optimal mesh refinement strategy can be learned by recurrent neural networks. This includes problems for which no optimal adaptive strategy is known yet

Recurrent neural networks as optimal mesh refinement strategies

We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input size, i.e., number of elements of the mesh. Moreover, for a general class of PDEs with solutions which are well-approximated by deep neural networks, we show that an optimal mesh refinement strategy can be learned by recurrent neural networks. This includes problems for which no optimal adaptive strategy is known yet