An Error Estimator for the Finite Element Approximation of Plane and Cylindrical AcousticWaves
J. E. Sebold, L. A. Lacerda and J. A. M. Carrer

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 106, No. 2, pp. 127-145, 2015
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Keywords Numerical Methods in Engineering, Finite Element Method, Helmholtz Equations, Plane and Cylindrical Wave Propagation.

This paper deals with a Finite Element Method (FEM) for the approximation of the Helmholtz equation for two dimensional problems. The acoustic boundary conditions are weakly posed and an auxiliary problem with homogeneous boundary conditions is defined. This auxiliary approach allows for the formulation of a general solution method. Second order finite elements are used along with a discretization parameter based on the fixed wave vector and the imposed error tolerance. An explicit formula is defined for the mesh size control parameter based on Padé approximant. A parametric analysis is conducted to validate the rectangular finite element approach and the mesh control parameter. The results of the examples show that the discrete dispersion relation (DDR) can be used for the rectangular finite element mesh refinement under predefined error tolerances. It is also shown that the numerical formulation is robust and can be extended to higher order finite element analyses.

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