New results on the a posteriori error analysis for Virtual Element Methods.

In order to describe the world and its phenomena, an essential concept is necessary: the derivative. Derivative is the mathematical way to describe the evolution in time or in space. By combining derivatives and some constraints observed, which are described as equations, the partial differential equations (PDE) arise. Most of the time we can just discuss some properties of the solution of a PDE, but finding the explicit solution can be analytically impossible. Numerical Analysis tries to solve this problem with the Finite Element Method (FEM). This method is based on finding an approximation of the real solution; it gets more precise as the degree of accuracy grows. FEM considers a discretization of the domain made by finite elements, defined by a triple which consists in the ‘geometrical shape’ E of the element forming the partition, a space of approximation functions living in E and a set of degrees of freedom. The main theme of the thesis focuses on the Virtual Element Method (VEM), a type of FEM, which has been introduced less than ten years ago. The thesis fits in the Adaptive Virtual Element Methods theory. It investigates the stabilizationfree a posteriori error analysis in polygonal meshes in 2d. The first novelty brought by this work stands in the extension of Beirão da Veiga et al. [2021] to the cases of triangular meshes with hanging nodes and polygons of higher degree. Assuming that any chain of recursively created hanging nodes is uniformly bounded, stabilization-free upper and lower bounds for the energy error are presented. The main difference with respect to the case of polynomials of degree one is that, given two triangles sharing an edge, the refinement of one of them brings some points to be both hanging nodes and proper nodes for the other triangle. On one hand, because of this, a re-definition of the hanging nodes is needed, on the other hand it simplifies a lot the proof of the Scaled Poincaré inequality. The second topic studied in the thesis is the extension of the analysis to the case of convex quadrangles. The main challenge here is the definition itself of the refinement. In the thesis the refinement consists in tracing the edges connecting the midpoints of two opposite edges of the quadrangles. In this way a quadrangle is reduced to four quadrangles. Also the space of polynomials of degree one has to be changed. Indeed, a polynomial of degree one is not uniquely determined by the value at the four vertices of the quadrangle. We then introduced a new functional space that contains the polynomials of degree one. Finally the enhanced version of this functional spaced and the stabilization-free a posteriori error analysis have been discussed.