Some Studies On Solving Convection-dominated Problems Based On SUPG Stabilized Method In 3D Anisotropic Meshes

Convection-diffusion equation is a kind of basic motion equation,which is an important kind of partial differential equation and widely used in many fields.How-ever,there are many obstacles when we solve the convection-dominated convection-diffusion equations,such as numerical oscillation.Therefore,the study of convection-dominant equations has received extensive attention.In this paper,we aim to study three-dimensional convection-dominant convection-diffusion equations with constant coefficients.The main work is as follows:First,we select SUPG as the stabilized method,the variational form of convection-diffusion equation based on this method is given.Meanwhile,we list some existed selections of stabilized parameters in anisotropic meshes appeared in present studies.Second,since we know that the discrete error of SUPG method is bounded by the sum of errors between the true solution and its interpolation function in different norms.In order to simplify the dependence of interpolation error on the solution,shape and size of the elements,the general solution is replaced by its second order Taylor expansion and consequently,the second derivative is expressed by the Hessian matrix of the solution.Then the solution is interpolated to obtain the prior error estimate of the true solution and its interpolation function in different norms under anisotropic meshes,so as to obtain the specific expression of the discrete error bound.Third,in order to generate the 3D anisotropic meshes,we construct the invertible affine map from general tetrahedral element to regular tetrahedral element,further calculate the discrete error expressions in the second part.Then,by minimizing the discrete error,the forms of the stabilization parameter and the monitor function are determined,then the monitor function is unified and the expression of the metric tensor is given.Fourth,in order to further optimize the derived stabilization parameter in the third part,under the assumption that the solution is isotropic,we consider respective-ly the convection-dominant and diffusion-dominant situations.By comparing with the selection of αk in DEE method,the coefficients of convection term and diffusion term in the stabilization parameters are settled finally,so that we obtain more prac-tical stabilization parameters.Finally,we compare the 3D results with the conclusions of 2D anisotropic meshes and find that they have the similar form.