A priori and a posteriori estimates of stabilized mixed finite volume methods for the incompressible flow arising in arteriosclerosis

Abstract The flow arising in arteriosclerosis is modeled by the incompressible flow with a slip boundary condition of friction type in this paper, whose weak solution satisfies a variational inequality. We develop and analyze the stabilized mixed finite volume methods for the resulting model. We first achieve the same optimal order results of the finite volume methods as those of the corresponding finite element methods under the same regularity on the exact solution and a slightly additional regularity on the source term. Also, a super-close result between the respective solutions of the finite element methods and finite volume methods is derived for the first time. For adaptivity, we further derive a posterior estimate for the present problem satisfying the variational inequality, which reduces the computational cost substantially and provides efficient error control for the solution. Finally, numerical results are shown to support the developed convergence theory.