Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

By Stefano Berrone, Andrea Borio. 2017

74 citations recorded by the source corpus.

In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the usual monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produces very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost.

Subfields: Computational Mechanics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/9b8b7a1d26534eca6216161db4de74b887b422dd

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In Computational Mechanics

1. Andrea Borio
2. Stefano Berrone

In Electrical and Electronic Engineering

1. Stefano Berrone
2. Andrea Borio

Citations in the corpus

• A residual a posteriori error estimate for the Virtual Element Method (2017, methodology): “We do not use larger values for the VEM order as the resulting linear systems turn to out to be too badly assembled due to the ill conditioning of the local projection matrices.(24) From Figure 4 we see the good agreement between the exact error and the a posteriori estimate (see also Table 1 for th”