A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian (Q6574916)

The authors consider a boundary value problem of a partial differential equation in a two-dimensional polygonal domain. This task represents a grad-div and curl-curl problem, which is related to a vector-valued Laplacian operator. A weak form of the problem is derived using a hybridization. Consequently, the authors arrange a nonconforming finite element method (FEM) based on a triangulation and polynomial elements. In [\textit{S. C. Brenner} et al., Numer. Math. 109, No. 4, 509--533 (2008; Zbl 1166.78006)], a problem of this type was solved using an FEM, which includes a penalty technique. Such a penalty parameter is also applied in the weak form. The aim is to obtain a convergent method of high order even in the case of corner singularities.\N\NThe authors prove existence and uniqueness of solutions of the weak form. Furthermore, stability estimates are shown, which are satisfied by the solution. Consequently, the authors deduce error estimates of the numerical solution in the \(L_2\)-norm and the energy norm, where the proof is based on weighted Sobolev spaces. These error estimates demonstrate the convergence of the FEM with arbitrary order provided that the exact solution is sufficiently regular and the polynomial degree is sufficiently large.\N\NFinally, the authors present results of numerical computations produced by their method, where an L-shaped domain is used to investigate corner singularities. The errors of the \(L_2\)-norm as well as the energy norm are depicted in tables. The observed convergence rates agree to the convergence rates predicted by the error analysis. In some cases, the observed convergence rates are slightly better than the expected convergence rates.