学术报告

报告题目：A penalized weak Galerkin spectral element methodfor second order elliptic equations

报告人：李会元  （中国科学院软件研究所研究员，教授）

报告时间：6月22日下午14:30—15:15

报告地点：数统院307

                                      数学与统计学院

                                        2019.6.20

报告摘要：A weakGalerkin spectral element method is proposed to solve second order partialdifferentialequations.  Following the idea of the weak Galerkin finite elementmethod, this method introduces  for the unknown solution a weak functionconsisting of one component on the elements together with one component on theelement interfaces, and replaces derivatives in the standard variational formwith weak derivatives defined on the space of weak functions on each element.As in a classic spectral element method, approximation spaces for weakfunctions on each triangular or parallelogram  element are defined fromorthogonal polynomials on the reference domain through a one-to-one mapping, and approximation spaces for weak derivatives are thenestablished via the Piola transform after an insight investigation of the weakgradient and the discrete weak gradient. To eliminate the effect of the possiblenullity of the discrete weak gradient and guarantee the wellposedness of theresulted algebraic system,  a penalty term defined on the edges issupplemented into the Galerkin approximation scheme. Error estimates for both the source problem and  the eigenvalue problem  on meshesconsisting of affine families of triangles and quadrilaterals are obtained inthe sequel, which are optimal in the mesh and size  suboptimal by one-half order withrespect to the polynomial degree.    

      Numericalexperiments for the eigenvalue problems are performed on both  the typicalsquare domain and L-shaped domain with  triangular meshes andquadrilateral meshes, which illustrate the effectiveness and high accuracy ofour penalized weak Galerkin spectral element method.