高波数Helmholtz 方程的内罚有限元方法

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-γ_r+iγ_i 的虚部 γ_i 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|γ_r|≤γ_i≤1, 那么存在与k,h,γ,R 无关的常数C₀;C₁;C₂, 使得当k³h²R ≤ C₀ 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/γ_i)RM(f, g), 其中M(f, g) := (‖f‖_L²(Ω) + R^-1/2‖g‖_L²(Γ)) + R^-1|g|_H^1/2(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iγ_>i 并令γ_i 趋于0+, 本文还在k³h²R ≤ C₀ 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.

Continuous interior penalty finite element methods for Helmholtz equation with high wave number

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 k^3h^2R ≤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.