研究目的
To truncate the Helmholtz scattering problem with high wave number using the PML technique and discretize it with CIP-FEM, proving stability, convergence, and preasymptotic error estimates.
研究成果
The truncated PML problem satisfies an inf-sup condition with constant O(k?1), and preasymptotic error estimates for CIP-FEM are derived as C1kh + C2k3h2. Numerical results confirm that tuning penalty parameters can significantly reduce pollution errors, making CIP-FEM superior to standard FEM for high wave number problems.
研究不足
The analysis assumes constant PML medium properties and specific mesh conditions (e.g., k3h2 sufficiently small). The method may not generalize to variable PML properties without modifications, and numerical tests are limited to 2D cases with equilateral triangulations.
1：Experimental Design and Method Selection:
The study uses the PML technique to truncate the Helmholtz equation and applies CIP-FEM for discretization. Theoretical models include inf-sup condition analysis and error estimation techniques.
2：Sample Selection and Data Sources:
The problem is defined in Rd (d=1,2,3) with a specific source function f, and numerical tests are conducted with varying wave numbers k and mesh sizes h.
3：List of Experimental Equipment and Materials:
Computational domain D, PML parameters (σ0, L, R), finite element meshes, and penalty parameters γe for CIP-FEM.
4：Experimental Procedures and Operational Workflow:
The PML is applied to truncate the domain, the variational formulation is set up, and CIP-FEM is implemented with numerical simulations performed in MATLAB.
5：Data Analysis Methods:
Error norms (H1 and L2) are computed, and results are compared with interpolation errors to assess pollution effects.
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