Research Activities

The Numerical Analysis Laboratory focuses its activities on the development and application of advanced numerical methods for interpolation, solving partial differential equations (PDEs), and function approximation. Main research topics include:

Multinode Shepard Methods: These methods provide rational approximations for interpolating large sets of scattered data, extending triangular Shepard interpolation. Applications include the numerical solution of PDEs such as the Poisson equation (with Dirichlet, Neumann, or mixed boundary conditions), the Black-Scholes equation for financial option pricing, and the approximation of functions with singularities.

Constrained Mock-Chebyshev Interpolation-Regression Method: This interpolation and regression technique aims to mitigate Runge's phenomenon by utilizing nodes derived from orthogonal polynomials. It has been successfully applied to quadrature problems, numerical differentiation, and integral equations.

Finite Element Enrichment: This research develops methods to enrich standard finite elements, enhancing their accuracy in solving PDEs over complex domains. The approach has been applied to triangular and simplicial finite elements.

Umbral Interpolation: Umbral interpolation is a general interpolation procedure dependent on a linear functional L and a delta operator Δ defined on a space of real-valued functions. A key feature is that, given L and Δ, the polynomial basis for interpolation can be written using a standardized matrix-determinant procedure. This interpolation procedure generates a class of quadrature formulas based on interpolation, utilizing derivatives of the integrand at specified nodes. Umbral interpolation has also been applied to solve differential problems and extended to the bivariate case.