# American Institute of Mathematical Sciences

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

 1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

* Corresponding author: Gang Bao

Received  July 2020 Revised  October 2020 Published  November 2020

Fund Project: 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

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.

Citation: Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020351
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A schematic of octree data structure
Two geometries to avoid hanging points. (left) Twin-tetrahedron geometry. (right) Four-tetrahedron geometry
Mesh refinement on the surface (red points are redefined midpoints on the boundary)
Example 1: (left) initial mesh on the $x_1 x_2$-plane. (right) adaptive mesh on the $x_1 x_2$-plane
Example 1: quasi-optimality of the a priori and a posteriori error estimates
Example 2: (left) an adaptively refined mesh with 63898 elements. (right) quasi-optimality of the a posteriori error estimate
 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.
 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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