Optimal Penalty Parameters For Interior Penalty Finite Element Methods For Helmholtz Equation On Triangular Meshes

When we use standard finite element method to solve Helmholtz equation with high wave number,the approximation efficiency decreases as the increase of the wave number of k,which is called pollution effect.And the pollution error and the phase difference(the difference between the k and discrete wave number of the finite element solution)are of the same order in H1 norm,they are all O(k2p+1h2p)where p is the order of finite element method.In this paper,we consider the selection of penalty parameter for interior penalty finite element methods for Helmholtz equation with high wave number on triangular meshes to reduce pollution error.Firstly,we consider the linear interior penalty finite element method on triangulations composed of isosceles right triangles and proved that we can reduce the phase difference of the penalty finite element solution to O(k5h4)through selecting penalty parameters.Secondly,for the linear interior penalty finite element method on triangulations composed of general isosceles triangles,we proved that the leading terms of the penalty parameters on the base and waist of the isosceles triangle should be inversely proportional to each side length’ square.Thirdly,we also consider quadratic interior penalty finite element method on equilateral triangulations.Penalty terms on jumps of the first and second normal derivatives at interior sides are introduced,respectively.We proved that selecting penalty parameters can also reduce the phase error from O(k5h4)of quadratic finite element method to O(k11h10).Finally,numerical experiments are provided to show that the pollution error of the finite element method can be reduced significantly by selecting the appropriate penalty parameters.