A Nonconforming Spectral Element Method Of Mixed Geometries Based On Duffy Mapping

The spectral-element method processes good computational properties,compared with the spectral method,such as its good applicability of tensor products,its mass ma-trix in the form of naturally diagonal matrices and suitability for parallel computation.However,the traditional spectral element computational domain is quadrilateral in normal conditions,which limits its its processing capacity adaptive computation in complex ge-ometries.Meanwhile,the computational efficiency of spectral element method in solving complex regions is limited by the consistent coordination of polynomial approximation orders.This article proposes a new idea which is based on the spectral method the triangular spectral-element method and noncomforming spectral-element method.The new method aims to implement nonconforming spectral-element method on the complex geometries of mixed elements,to enable the new method to encrypt locally and hence strengthen its processing capacity in complex geometries.This article uses the Duffy Mapping to map the triangle to t.he square,high oreder polynomial approximation method in triangular regions and lower order polynomial ap-proximation method in quadrangular regions.High order matching are used in interface while nodal basis functions are in the forms of tensor products.Further,we ran the analysis on well-posedness and convergence of this method and prove its feasibility by numerical experiments.The rational basis functions in the triangle area are constructed through the Duffy mapping in the reference square.We can construct a nodal basis func-tion through the tensor products.We use high-order polynomials for approximation in triangular areas,and use lower-order polynomials for quadrilateral areas.We will prove the existence,uniqueness and convergence of soliutions.Then we will carry out some numerical experiments to demonstrate the efficiency of this method.