An O(p<SUP>3</SUP>) <i>hp</i>-version FEM in two dimensions: Preconditioning and post-processing

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dc.contributor.authorAinsworth, Mark
dc.contributor.authorJiang, Shuai
dc.contributor.authorSanchez, Manuel A.
dc.date.accessioned2025-01-23T21:14:21Z
dc.date.available2025-01-23T21:14:21Z
dc.date.issued2019
dc.description.abstractThe 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.
dc.description.abstractThe construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p, at a cost of O(p(3)) operations, is one of the main contributions of the present work. The preconditioner is based on an abstract Additive Schwarz Method recently developed by the authors. The preconditioner can be implemented at a cost of O(p(3)) operations by exploiting properties of the Bernstein polynomials. In particular, we present an algorithm which allows one to invert the interior block of the element mass matrix in O(p(3)) operations. Numerical examples are provided to illustrate the applicability of the Bernstein basis to challenging non-linear reaction-diffusion problems, non-linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers. (C) 2019 Elsevier B.V. All rights reserved.
dc.description.funderDepartment of Defense (DoD), United States through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program
dc.fuente.origenWOS
dc.identifier.doi10.1016/j.cma.2019.03.020
dc.identifier.eissn1879-2138
dc.identifier.issn0045-7825
dc.identifier.urihttps://doi.org/10.1016/j.cma.2019.03.020
dc.identifier.urihttps://repositorio.uc.cl/handle/11534/101034
dc.identifier.wosidWOS:000468163500032
dc.language.isoen
dc.pagina.final802
dc.pagina.inicio766
dc.revistaComputer methods in applied mechanics and engineering
dc.rightsacceso restringido
dc.subjectPreconditioning mass matrix
dc.subjectHigh order finite elements
dc.subjectPost-processing
dc.titleAn O(p<SUP>3</SUP>) <i>hp</i>-version FEM in two dimensions: Preconditioning and post-processing
dc.typeartículo
dc.volumen350
sipa.indexWOS
sipa.trazabilidadWOS;2025-01-12
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