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An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

  • * Corresponding author: Gang Bao

    * Corresponding author: Gang Bao 

The first author is supported in part by an NSFC Innovative Group Fund (No.11621101). The last author is supported in part by the NSF grant DMS-1912704

Abstract / Introduction Full Text(HTML) Figure(6) / Table(1) Related Papers Cited by
  • This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

    Mathematics Subject Classification: Primary: 65N30, 78A45, Secondary: 35Q60, 78M10.

    Citation:

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  • Figure 1.  A schematic of octree data structure

    Figure 2.  Two geometries to avoid hanging points. (left) Twin-tetrahedron geometry. (right) Four-tetrahedron geometry

    Figure 3.  Mesh refinement on the surface (red points are redefined midpoints on the boundary)

    Figure 4.  Example 1: (left) initial mesh on the $ x_1 x_2 $-plane. (right) adaptive mesh on the $ x_1 x_2 $-plane

    Figure 5.  Example 1: quasi-optimality of the a priori and a posteriori error estimates

    Figure 6.  Example 2: (left) an adaptively refined mesh with 63898 elements. (right) quasi-optimality of the a posteriori error estimate

    Table 1.  The adaptive FEM-DtN algorithm

    1 Given a tolerance $ \varepsilon > 0 $;
    2 Choose $ R $, $ R' $ and $ N $ such that $ \varepsilon_{N}<10^{-8} $;
    3 Construct an initial tetrahedral partition $ \mathcal{M}_h $ over $ \Omega $ and compute error estimators;
    4 While $ \eta_K>\varepsilon $, do
    5     mark $ K $, refine $ \mathcal{M}_h $, and obtain a new mesh $ \hat{\mathcal{M}}_h $.
    6     solve the discrete problem on the $ \hat{\mathcal{M}}_h $.
    7     compute the corresponding error estimators;
    8 End while.
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