10 research outputs found
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