Introduction Of Support Operator Method And The Comparison Of SOM And MFEM

We derive a mimetic finite di?erence method arise recently: supportoperator method (SOM). The establishment of the method based on thebasic mathematical principle. It has maintained some good physical prop-erties, such as conservative and symmetrical. On solving di?usion problemsin strongly heterogeneous anisotropic materials with distorted grids, It hasstrong vitality. When the grid is seriously distorted, or even concave, themethod also has good stability.First we consider the following elliptic boundary value problems:Because the mimetic FDM'Ms are based on discrete analogs of first-order coordinate-invariant operators , it is natural to write the equationas a system of first order systems:By introducing the inner product, we can easily find that:The major steps of SOM is making operator G,div discrete. In chap-ter one, we brie?y introduce discrete methods of the first order di?erentialoperator: first is by the coordinate invariable definition. for example: thedivergence operator is discrete based on the following definition : Another approach is viewed the operator which we want to be discreteas the conjugator of which has been discrete. For example, If we want todiscrete the gradient operator G. If we have obtained the discrete diver-gence operator and known the G = div? and (u,divW?→)H?(W?→,G u)H = 0.Now we can obtain discrete gradient indirectly.In SOM, we choose divergence div as the prime operator, In orderto obtain discrete gradient , we introduced two kinds of inner products,natural inner products and formal inner products. And also, we establishthe relationship between the two inner products by operators.In this paper, we interpret the SOM detailedly in logic quadrilateralmesh and non-structural triangular grid , including the definition of dis-crete function space and the inner product in it, the choice of weights, andso on.The traditional cell-centered support-operators method leads to adense di?usion matrix on non-orthogonal grids. Here we introduce a newvariant of the cell-centered support-operators method which always leadsto a local di?usion stencil at the expense of additional side-center pressureunknows. Her we refer to this new methods as a"local"support operatorsmethod.The error estimate of SOM come through a long period. Until 2001,someone proved convergence of the SOM for linear di?usion by first de-veloping a connection of this mimetic discretization with Mixed FiniteElement(MFEM) methods. As we can see, the SOM discrete is at least ofthe same order of accuracy as the lowest order RT finite element method.We also make a comparison between SOM and MFEM in the part ofnumerical examples. we can find that in cases of quadrilateral mesh, nomatter how the regularization of the mesh is, the MFEM is better thanthe SOM in the accuracy and the order, but in the perspective of computeprice, the SOM is lower;however, in the cases of triangular mesh, the SOMis better in the perspective of order and compute price, especially whenthe grid is severe distorted, the di?usion coe?cient is discontinuous, theadvantage is more obvious.