Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Orthogonal polynomial bases in the Mixed Virtual Element Method

Stefano Berrone, Stefano Scialò, Gioana Teora

TL;DR

An orthogonal vector-polynomial basis is introduced which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case.

Abstract

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures.

Orthogonal polynomial bases in the Mixed Virtual Element Method

TL;DR

An orthogonal vector-polynomial basis is introduced which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case.

Abstract

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures.
Paper Structure (18 sections, 1 theorem, 104 equations, 9 figures, 3 tables)

This paper contains 18 sections, 1 theorem, 104 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The continuous problem and the mixed variational formulation
  3. The Mixed Virtual Element Method
  4. The discrete mixed variational formulation
  5. Polynomial basis
  6. Polynomial bases for the polynomial space
  7. A full L2E-orthonormal approach
  8. Some implementation details
  9. Term which includes divergence
  10. Computation of L2E-projection of basis functions of VEM space for the velocity variable
  11. The diffusion term
  12. The advection term
  13. The reaction term
  14. Numerical experiments
  15. Test1: Convergence rates
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $p_h$ the solution to (eq:discreteMixedVariationalFomrulation) and let $p_I \in Q_h$ be the interpolant of $p$. Then, for $h$ sufficiently small,

Figures (9)

  • Figure 1: Test1: Behaviour of errors \ref{['eq:L2pressure']}, \ref{['eq:L2velocity']} and \ref{['eq:superconvergence']} at varying $k$ on square meshes. Pictures on each row refer to a different mesh refinement level: 100, 400 and 1600 element meshes from top to bottom.
  • Figure 2: Test2: Maximum condition number of local matrices among elements, at varying $k$. Mesh with aspect ratio 10.
  • Figure 3: Test2: Maximum condition number of local matrices among elements at varying $k$. Mesh with aspect ratio 50.
  • Figure 4: Test2: Maximum condition number of local matrices among elements at varying $k$. Mesh with aspect ratio 100.
  • Figure 5: Test2: Behaviour of errors \ref{['eq:L2pressure']}, \ref{['eq:L2velocity']} and \ref{['eq:superconvergence']} at varying $k$ on rectangular meshes. Each row represents a different mesh: 10, 50, 100 from top to bottom.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 3.1: Superconvergence result
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3