We introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the Stokes equations. One pair has each component of its velocity field to be approximated by the $P_1$ nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two-dimensional subspaces removed: one removed space is due to the integral mean--zero property and the other space consists of global checker--board patterns. The other pair consists of the velocity space as the $P_1$ nonconforming quadrilateral element enriched by a globally one--dimensional macro bubble function space based on $DSSY$ (Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure field is approximated by the piecewise constant function with mean--zero space eliminated. We show that two element pairs satisfy the discrete inf-sup condition uniformly. And we investigate the relationship between them. Several numerical examples are shown to confirm the efficiency and reliability of the proposed methods

On adaptively refined quadrilateral or hexahedral meshes, one usually employs constraints on degrees of freedom to deal with hanging nodes. How these constraints are constructed is relatively straightforward for conforming finite element methods: The constraints are used to ensure that the discrete solution space remains a subspace of the continuous space. On the other hand, for nonconforming methods, this guiding principle is not available and one needs other ways of ensuring that the discrete space has desirable properties. In this paper, we investigate how one would construct hanging node constraints for nonconforming elements, using the Douglas-Santos-Sheen-Ye (DSSY) element as a prototypical case. We identify three possible strategies, two of which lead to provably convergent schemes with different properties. For both of these, we show that the structure of the constraints differs qualitatively from the way constraints are usually dealt with in the conforming case.

The P1$$ {P}_1 $$–nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non‐invertible matrices. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensions is provided.

A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided.

Solar power is an important renewable energy resource that plays a pivotal role in replacing fossil fuel generators and lowering carbon emissions. Since sunlight, which is highly dependent on meteorological factors, is highly volatile, the difficulty in collecting real-time data from renewable energy power plants poses a major threat to maintaining the stability of the entire power system in the target area. A high-performance wireless metering modem is required to monitor the renewable energy generation power of the entire target area in real-time. However, installing such devices on all sites is expensive, so we propose a system that uses deep learning to estimate the generation power of a target site based on the power generations of some sample sites. We use clustering and distance-based sampling to extract a sample site corresponding to each target site and use the recurrent neural network (RNN)-based attention techniques to estimate the generation of target sites from the sample sites. Our experiments show that the proposed RNN-based attention models significantly improve estimation accuracy compared to the baseline model or other deep learning models, irrespective of the number or location of sample sites.INDEX TERMS deep learning, real-time estimation, attention, long short-term memory, solar power generation estimation.