Stability and convergence of spatial discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics equations

Abstract

In this paper, the stability and convergence results of finite volume method for the time-dependent incompressible magnetohydrodynamics equations are developed based on triangulation. The linear polynomials are used to approximate the velocity, magnetic fields and pressure. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection method is employed. Firstly, two self-adjoint linear mappings are introduced to establish the equivalences of temporal terms and bilinear terms in finite element method and finite volume method. Secondly, the existence and uniqueness of numerical solutions are presented by using the Oseen iteration and mathematical induction method. Thirdly, H2-stability and optimal error estimates are provided by introducing the Galerkin projection of generalized bilinear terms in finite volume method, using the energy method and constructing the corresponding dual problem. The main theoretical findings of this paper are as follows:‖(uh(t),Hh(t))‖02+∫0t(min⁡{ν,γσ−1}‖(uh,Hh)‖12+G(ph,ph))ds≤C,min⁡{ν,γσ−1}‖(uh(t),Hh(t))‖12+G(ph(t),ph(t))+∫0t‖(uht,Hht)‖02ds≤C,‖∇(uh,Hh)‖02+∫0tmin⁡{ν,γσ−1}‖(A1huh,A2hHh)‖02ds≤C,τ12(t)‖(u(t)−uh(t),H(t)−Hh(t))‖0+h(‖(u(t)−uh(t),H(t)−Hh(t))‖1+τ12(t)‖p(t)−ph(t)‖0)≤Ch2, where τ(t)=min⁡{1,t}, the letters ν,σ are the physical parameters, C and γ are two positive constants independent of the mesh size h, (u,H,p) and (uh,Hh,ph) are the solutions of problems (2.1) and (4.1). Finally, some numerical results are presented to verify the established theoretical findings and show the performances of the considered numerical method.