| Finite volume element method is an important numerical method for solving partial differential equations.The main advantage of this method is that it could maintain the conservation of corresponding physical quantities.In this paper,we study some applications of the finite volume element method,which include the theoretical analysis of two finite volume element schemes for the incompressible Stokes equations,numerical simulation of moving interface problems using the finite volume element method in ALE framework,and monotonicity correction for second order element finite volume methods of anisotropic diffusion problems.Firstly,for the incompressible Stokes problem,we study the finite volume method based on MINI element pair.The trial spaces of velocity and pressure are taken as MINI element space and linear element space,respectively,and the test spaces are the piecewise constant space corresponding to the respective dual partition.The corresponding saddle point problem is asymmetric.With the help of two new transformation operators,we establish the equivalence of bilinear forms for gradient operator between finite volume methods and finite element methods,and the equivalence of bilinear forms for divergence operator between finite volume methods and finite element methods.Furthermore,we analyze the stability and convergence of the scheme.Numerical experiments confirmed the theoretical results.Secondly,we consider the Taylor-Hood element finite volume method for incompressible Stokes problems.The trial space of velocity is taken as a quadratic finite element space,and the test space is the piecewise constant space on the optimal dual partition.The trial space and the test space of pressure are both linear element space.Similarly,the corresponding saddle point problem is asymmetric.By constructing a new transformation operator,the equivalence between the bilinear form of gradient operator and the corresponding bilinear form in finite element methods is established.And then the stability and convergence are obtained.Numerical experiments confirmed the theoretical results.Thirdly,we construct the finite volume element method in ALE framework for the Stokes/Parabolic moving interfaces problem.We discretize the Stokes equation on one side of the interface using MINI element finite volume method,and discretize the parabolic equation on the other side of the interface using linear element finite volume method.Then we obtain the full discretization by using the backward Euler scheme in the time direction.The numerical results show that the finite volume element method in ALE framework can effectively simulate the moving interface problem,and the optimal convergence order is obtained.Fourthly,we consider the monotone correction of second order element finite volume schemes,such that the corrected scheme is monotone,where monotonicity means positivity preserving.The main idea of monotonic correction are as follows: we regard the line integrals on the boundaries of dual elements as numerical fluxes.After separating the main part,which is a two-point flux with positive coefficients,from the numerical flux,we nonlinearly embed rest part into the positive and negative coefficients of the main part according to the sign of the rest part.Thereby a nonlinear monotone second-order element finite volume scheme is obtained.Furthermore,for the time-dependent problems,we also propose a monotone correction method for the time derivative term.Numerical experiments show that the corrected schemes have monotonicity and maintain the convergence order of the original schemes. |
