Quasi-optimal nonconforming methods for symmetric elliptic problems. I -- Abstract theory
For researchers in numerical analysis, this provides a theoretical framework for designing and analyzing nonconforming finite element methods, but it is a theoretical contribution without concrete numerical results.
The paper develops an abstract theory characterizing quasi-optimality of nonconforming methods for symmetric elliptic problems, determining the quasi-optimality constant and quantifying the impact of nonconformity via two consistency measures. It shows that constructing quasi-optimal methods reduces to choosing smoothing operators that map discrete functions to conforming ones.
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces.