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Nitsche's Method For Helmholtz Problems with Embedded Interfaces.

Zilong Zou¹, Wilkins Aquino¹, Isaac Harari²

¹Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, U.S.A.

International Journal for Numerical Methods in Engineering

|July 18, 2017

Summary

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This study introduces Nitsche's formulation for embedded interfaces in Helmholtz problems, simplifying complex geometries. The method ensures well-posedness and convergence for finite element analysis, confirmed by plane-wave examples.

Area of Science:

Computational Mathematics
Numerical Analysis
Partial Differential Equations

Background:

Discretization of complex geometries in Helmholtz problems is challenging.
Embedded interfaces simplify mesh generation for intricate material boundaries.
Weak enforcement of kinematic constraints is needed for embedded interfaces.

Purpose of the Study:

To develop and analyze Nitsche's formulation for enforcing kinematic constraints at embedded interfaces in Helmholtz problems.
To establish the well-posedness and finite element convergence for these problems.
To validate the analytical findings with numerical examples.

Main Methods:

Application of Nitsche's method for weak enforcement of kinematic constraints.
Analytical investigation of well-posedness and inf-sup stability.

Keywords:

Embedded interface Helmholtz problem Nitsche’s method

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Finite element discretization and numerical simulations of 2D plane-wave problems.

Main Results:

Analytical results confirm the well-posedness of the Helmholtz variational problems.
Convergence of finite element discretizations is established with judicious choice of the Nitsche's stabilization parameter.
Numerical examples demonstrate asymptotic convergence and validate the theoretical analysis.

Conclusions:

Nitsche's formulation provides a robust method for handling embedded interfaces in Helmholtz problems.
The proposed approach simplifies discretization while maintaining analytical rigor.
The findings pave the way for efficient numerical solutions in complex geometric scenarios.