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In this paper the optimal L 2 error estimates of the finite volume element methods (FVEM) for Poisson equation are discussed on quadrilateral meshes. The trial function space is taken as isoparametric bilinear finite element space on quadrilateral partition, and the test function space is defined as piecewise constant space on dual partition. Under the assumption that all elements on quadrilateral meshes are O(h 2 ) quasi-parallel quadrilateral elements, we prove convergence rate to be O(h 2 ) in L 2 norm.Keywords Finite volume element methods · L 2 error estimate · Quadrilateral meshes · Isoparametric bilinear element · Dual partition · Quasi-parallel quadrilateral Mathematics Subject Classifications (2000) 65N30 · 65N15Communicated by Zhongying Chen.

SUMMARY A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.

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This paper is concerned with the finite volume element methods on quadrilateral mesh for second-order elliptic equation with variable coefficients. An error estimate in L 2 norm is shown on the quadrilateral meshes consisting of h 2 -parallelograms. Superconvergence of numerical solution is also derived in an average gradient norm on h 2 -uniform quadrilateral meshes. Numerical examples confirm our theoretical conclusions.

SUMMARYNonequilibrium radiation diffusion problems are described by the coupled radiation diffusion and material conduction equations. Because of the highly nonlinear, strong discontinuous, and tightly coupled phenomena, solving this kind of problems is a challenge. We construct two finite volume element schemes for the equations. One of them is monotone on many kinds of meshes, which is proved theoretically and verified by numerical tests. The other one is hard to satisfy the monotonicity, but this defect can be corrected by different repair techniques. Numerical results show that these new methods are practical and efficient on distorted meshes.Copyright © 2013 John Wiley & Sons, Ltd.

We consider the precondition of linear systems which resulted from the finite volume method (FVM) for elliptic boundary value problems. With the help of the interpolation operator from the trial space to the test space of the FVM and the operator induced by the FVM bilinear form, we show that both wavelet preconditioners and multilevel preconditioners designed originally for the finite element method (FEM) of a boundary value problem can be used to precondition the FVM of the same boundary value problem. We prove that such preconditioners ensure that the resulting coefficient matrix of the FVM has a uniformly bounded condition number. We present seven numerical examples to confirm our theoretical findings.

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