Abstract
The paper discusses various formulations of the recently developed higher order Multipoint Meshless Finite Difference Method. The novel multipoint approach is based on raising the order of approximation of the unknown function by introducing additional degrees of freedom in stencil nodes, taking into account e.g. the right hand side of the considered differential equation. It improves the finite difference solution without increasing the number of nodes in an arbitrary irregular mesh. In general, the standard version of the Meshless (Generalized) FDM is based on the strong problem formulation. The extensions of the multipoint meshless FDM allow for analysis of boundary value problems posed in various weak formulations, including variational ones (Galerkin, Petrov-Galerkin), minimization of the energy functional, and the meshless local Petrov-Galerkin. Several versions of the multipoint method are proposed and examined. The paper is illustrated with some examples of the multipoint numerical tests carried out for the weak formulations and their comparison with those obtained for the strong formulation.
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Jaworska, I. (2023). On the Weak Formulations of the Multipoint Meshless FDM. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds) Computational Science – ICCS 2023. ICCS 2023. Lecture Notes in Computer Science, vol 10476. Springer, Cham. https://doi.org/10.1007/978-3-031-36027-5_41
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