This thesis introduces a higher-order numerical method for elliptic boundary value problems. The discretization method belongs to the class of mimetic discretizations, which translate as many of properties of the continuous problem to the discrete system, aiming to improve accuracy and reliability. The novelty lies in the application of B-splines as basis functions in a dual grid approach. B-splines or basis splines are piecewise polynomials with a certain degree of continuity between the polynomial pieces. Therefore, splines offer an attractive compromise between piecewise linear functions, commonly seen in finite element analysis, and the Lagrange polynomials from spectral element methods.