Application And Research Of HDG Method Based On Hybrid Meshes

It is very important to find a fast and efficient method to solve partial differential equations,which promote the development of science.For example,computational electromagnetics,which can be transformed into Maxwell equations by mathematical theory,is of great importance in many fields,such as engineering and aerospace science,communication and so on.The application of electromagnetics has also promoted the development of the computational techniques for solving the Maxwell equations.In this paper,we first introduce some effective methods applied to solve Maxwell equations,such as the finite element method,discontinuous Galerkin(DG)method and so on.On the basis of these methods,people developed a hybrid discontinuous Galerkin(HDG)method.It is based on the introduction of the hybrid term,which leads to the formation of a linear system with only the hybrid terms.This work is to use HDG method solving the two-dimensional time-harmonic Maxwell equations.The HDG method has all advantages of the DG method,and at the same time it reduces the number of global coupling degrees of freedom(DOFs),making the solution solving more quickly.However,the general HDG method based on structured mesh is only applicable to the shape of the rules of graphics,graphics for complex shapes are difficult to adapt.Unstructured mesh is a good solution to this problem,but it consumes a lot of computing resources.The hybrid mesh technology which based on structured mesh and unstructured mesh has emerged,which represents the development trend of mesh technology.In this paper,we use the hybrid-mesh HDG method which based on the combination of unstructured triangle and structured quadrilateral to solve the two-dimensional time-harmonic Maxwell equation.During the process of mesh generation,the area is divided into two parts,the inner domain with complex geometry is discretized into the unstructured triangle mesh,and the rest external free space is discretized into structured quadrilateral mesh.In the HDG framework,the interface of the two meshes can be well coupled.The HDG method based on hybrid mesh keep all its advantages such as adaptive complex geometries,easy to obtain high accuracy,hp-adaptive and natural parallelism,and it reduces the computational complexity andsize of the problem so that we can obtain a smaller linear system.At the same time,hybrid mesh combines the advantages of flexibility and completeness of structured mesh,being suitable for complex shape,mesh adaptive and other advantages of unstructured mesh.And it not only meets the application of complex shapes,but also overcomes the shortcomings of the large consumption of computing resources.By the end of paper,we use the ordinary electromagnetic field scattering model and nanowires scattering model for numerical simulation experiment to implement this method,and compare it with general unstructured mesh HDG.Under the same accuracy,the HDG method based on hybrid mesh has less degrees of freedom,computing time and computing resources.The results prove that the superiority of the HDG method based on hybrid mesh for solving Maxwell equations.