【摘要】 本文考虑二维和三维区域上高波数Helmholtz散射问题的线性内罚有限元方法.该散射问题的边界条件取为一阶吸收边界条件.本文证明了,如果加罚参数γ=γr+iγi的虚部γi大于零,那么内罚有限元方法是绝对稳定的,即对任意k,h,R>0都存在唯一解.这里k是波数,h为网格尺寸,R是区域的直径.进一步地,如果|γr|γi1,那么存在与k,h,γ,R无关的常数C0,C1,C2,使得当k3h2RC0时,该方法的H1误差界为(C1kh+C2k3h2R)RM(f,g),当k3h2R>C0且kh有界时,H1误差界为(C1kh+C2/γi)RM(f,g),其中M(f,g):=(∥f∥L2(Ω)+R-1/2∥g∥L2(Γ))+R-1|g|H1/2(Γ).另外,本文还推导了L2误差估计.注意到γ=0时内罚有限元方法就是经典的有限元方法,通过取加罚参数为iγi并令γi趋于0+,本文还在k3h2RC0的条件下,得到了有限元方法的稳定性和误差估计.作者以前的工作只考虑了加罚参数为纯虚数的情形并且没有考虑对R的依赖关系.更多还原

【Abstract】 This paper analyzes some continuous interior penalty finite element method(CIP-FEM) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions.It is proved that if the penalty parameter is chosen as a complex number with positive imaginary part γ = γ r + iγ i,then the CIP-FEM is absolute stable,i.e.,well-possed for any k,h,R > 0,where k is the wave number,h is the mesh size,and R is the diameter of the domain.Furthermore,if |γ r | γ i 1,then there exist constants C0,C1,C2 independent of k,h,γ,R,such that the H 1 error is bounded by(C1kh + C2k3h2 R)R M(f,g) when k 3 h 2 R C 0,and by(C1kh +C2/γi)R M(f,g) when k3h2R > C 0 and kh 1,where M(f,g):=(∥f∥L2(Ω) + R-1/2 ∥g∥ L2(Γ)) + R1 |g| H1/2(Γ).Optimal order L2 error estimates are also derived.By taking γ = iγ i and letting γ i → 0+ in the CIP-FEM,stability and error estimates are obtained for the standard FEM under the condition k3h2RC0.The previous work of the author considered only pure imaginary penalty parameters and did not consider the dependence on R.更多还原