A projection-based stabilized virtual element method for the unsteady incompressible Brinkman equations

Highlights：

• Compared with “residual stabilization term” which contains the projection of the second derivative and additional coupling terms, the projection-based stabilization term brings simpler schemes and less amount of calculation.

• The error analysis is discussed in two cases, when the viscosity coefficient μ>0 is not too small, and when μ is small (close to 0). The generic constant C is independent of μ.

• We shows the convergence order of the two cases (μ=1 and μ=1e−8) in numerical example 1. The numerical results agree with the theoretical analysis. Additionally, we make comparison with other schemes in numerical example 2.

• This paper is a basis for the construction of a modified formulation which is uniformly convergent with respect to μ. To some extent, it can be viewed as the first step for a series of works.

摘要

•The velocity and the pressure variable of the unsteady Brinkman equation are both approximated in H1 virtual element spaces. Projection-based stabilization term is introduced to make the formulation keep stable with spaces containing “equal-order” polynomials.•Compared with “residual stabilization term” which contains the projection of the second derivative and additional coupling terms, the projection-based stabilization term brings simpler schemes and less amount of calculation.•The error analysis is discussed in two cases, when the viscosity coefficient μ>0 is not too small, and when μ is small (close to 0). The generic constant C is independent of μ.•We shows the convergence order of the two cases (μ=1 and μ=1e−8) in numerical example 1. The numerical results agree with the theoretical analysis. Additionally, we make comparison with other schemes in numerical example 2.•This paper is a basis for the construction of a modified formulation which is uniformly convergent with respect to μ. To some extent, it can be viewed as the first step for a series of works.