[Paper Review] A Unified Analysis of Nonconforming Virtual Element Methods for Convection Diffusion Reaction Problem

Dibyendu Adak, E. Natarajan|arXiv (Cornell University)|Jan 6, 2016

Advanced Numerical Methods in Computational Mathematics5 references3 citations

TL;DR

This paper introduces a stabilized, nonconforming virtual element method (VEM) for convection-diffusion-reaction problems with dominant convection, using an $L^2$-projection operator to ensure stability and polynomial consistency. The framework achieves optimal convergence in the limit of vanishing diffusion and enables computation via degrees of freedom without explicit basis functions, though convergence requires $k \geq 2$ polynomial degree and higher source regularity.

ABSTRACT

We discuss nonconforming virtual element method for convection dominated (diffusive coefficient is very small compared to convective coefficient and reac- tion coefficient ) convection-diffusion-reaction equation using L^2 projection operator.In this paper we stabilize the stabilization terms using same technique which is used for stabilization of symmetric part in VEM, where T is an arbitrary element, and assume H^2(T) regularity on each element to prove polynomial consistency where v_h is approximate solution.We have shown that linear nonconforming VE is not convergent for convection dominated convection-diffusion reaction problem and higher regu larity of f ,source term is also needed for convergence analysis.The novelty of this paper is we introduce a new SDFEM type nonconforming virtual element method for convection-dominated convection diffusion reaction equation, and discuss the computability issue using degrees of freedom of element without explicit knowledge of basis functions of virtual element methods.The present framework is stable in the limit of vanishing diffusion.

Motivation & Objective

• To develop a stable, nonconforming virtual element method for convection-diffusion-reaction problems with small diffusion.
• To address non-physical oscillations in standard Galerkin methods via streamline diffusion-type stabilization.
• To ensure polynomial consistency and stability using $L^2$-projection and higher regularity assumptions on the discrete solution.
• To establish computability of the method using only degrees of freedom, avoiding explicit knowledge of basis functions.
• To prove convergence and a priori error estimates under minimal assumptions on the source term and mesh.

Proposed method

• The method employs an $L^2$-projection operator to stabilize the non-symmetric bilinear form arising from convection and reaction terms.
• Stabilization terms are constructed using the same technique as for symmetric parts in VEM, specifically $\int_T (\vec{b} \cdot \nabla u)(\vec{b} \cdot \nabla v)$.
• Polynomial consistency is proven under the assumption that $v_h|_T \in H^2(T)$, with higher regularity $H^3(T)$ required for convergence.
• The discrete bilinear form is analyzed using a mesh-dependent norm to establish coercivity and boundedness.
• Computational procedures rely on degrees of freedom to compute moments and projections, avoiding explicit basis functions.
• The framework uses elliptic projection operators to compute $L^2$-projections for polynomials up to degree $k-2$, with extensions for higher degrees via approximation.

Experimental results

Research questions

• RQ1 Can a nonconforming virtual element method be stabilized for convection-dominated convection-diffusion-reaction problems using $L^2$-projection?
• RQ2 Is polynomial consistency achievable in nonconforming VEM for non-symmetric problems with $L^2$-projection?
• RQ3 Does the method remain stable and convergent in the limit of vanishing diffusion ($\epsilon \to 0$)?
• RQ4 Can all terms in the discrete bilinear form be computed using only degrees of freedom, without explicit basis functions?
• RQ5 What regularity assumptions are required for convergence, and does linear nonconforming VEM fail in this context?

Key findings

• Linear nonconforming virtual elements do not converge for convection-dominated problems, requiring $k \geq 2$ polynomial degree for convergence.
• The method achieves optimal convergence rates under the assumption that the source term $f$ belongs to $H^s(T)$ with $s \geq 1$ on each element $T$.
• The $L^2$-projection operator enables stable and consistent analysis for non-symmetric bilinear forms, even in the limit of small diffusion.
• All terms in the discrete bilinear form are computable using only degrees of freedom, confirming the practical feasibility of the method.
• The framework is stable and coercive in a mesh-dependent norm, ensuring robustness for convection-dominated problems.
• The consistency error is bounded via patch tests, accounting for nonconformity along interior edges.