Browsing by Author "Jiang, Shuai"

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    An O(p3) hp-version FEM in two dimensions: Preconditioning and post-processing
    (2019) Ainsworth, Mark; Jiang, Shuai; Sanchez, Manuel A.
    The Bernstein polynomials have been known for over a century and are widely used in the spline literature, computer aided geometric design, and computer graphics. However, the realization that the Bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development. For instance, it is known (Ainsworth et al., 2011) that the Bernstein basis can be exploited to compute all of the entries in the load vector in O(p(3)) operations even in the case of non-linear problems on curvilinear elements for a degree p approximation. Moreover, the element matrices can be assembled in O(1) operations per entry. We show that properties of the Bernstein polynomials can also be exploited to obtain O(p(3)) complexity procedures for all of the main components needed to implement a high order finite element code including: computation of the residuals needed for an iterative solution method; evaluating the action of a preconditioner for the global mass matrices; and, visualization and post-processing of the resulting finite element approximations.