Estimation of A Posteriori Computational Error by the Higher Order Multipoint Meshless FDM

The main objective of this paper is to present a possibility of high quality a posteriori error evaluation based on reference solutions obtained by means of the new multipoint meshless finite difference method. Due to its higher order approximation, the multipoint results may be used as improved reference solution instead of the true analytical one to estimate the errors. Several types of a posteriori error estimators which can be used to evaluate the calculation error are described here. The results of selected numerical benchmarks are considered

Stencil and kernel optimisation for mesh-free very high-order generalised finite difference method

Generalised Finite Difference Methods and similar mesh-free methods (Point set method, Multipoint method) are based on three main ingredients: a stencil around the reference node, a polynomial reconstruction and a weighted functional to provide the relation sbetween the derivatives at the reference node and the nodes of the stencil.Very few studies were dedicated to the optimal choice of the stencil together with the other parameters that could reduce the global conditioning of the system and bring more stability and better accuracy. We propose a detailed construction of the very high-order polynomial representation and define a functional that assesses the quality of the reconstruction. We propose and implement several techniques of optimisation and demonstrate the advantages in terms of accuracy and stability.J. Figueiredo acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UID/FIS/04650/2019. S. Clain the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional , through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UIDB/00324/2020