Convergence study of the finite element method for elastic waves in anisotropic media

Smirnov I.E.
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The paper presents a detailed and systematic convergence analysis of the finite element method for modeling elastic wave propagation in anisotropic media possessing orthotropic symmetry of the elasticity tensor. Spatial discretization is based on the application of the finite element method with quadratic basis functions $P_2$ on unstructured tetrahedral meshes, which provides flexibility in handling complex geometries. For time integration, the Newmark scheme with parameters $\beta = 0.302$, $\gamma = 0.6$ is used; however, the choice of this particular scheme does not affect the spatial convergence, which is the subject of the present study.

The convergence analysis is performed on a benchmark problem with a smooth localized initial pulse in a half-space with free-surface boundary conditions. The initial conditions are chosen such that the solution remains smooth and the pulse does not reach the boundaries of the computational domain until the moment of solution comparison. This allows one to eliminate the influence of boundary condition approximation on the studied convergence and to focus exclusively on the spatial component of the error. A series of computational calculations is carried out on six successively refined unstructured meshes with the number of tetrahedra ranging from 39 728 to 372 904. The comparison is performed against a reference solution obtained on a substantially finer mesh, which ensures sufficient accuracy of the reference data. To correctly compute the norm of the difference, the solutions on all meshes were projected onto a common coarse spatial mesh.

It is established that the convergence order in the $L_2$ norm is 2.314 when using the median edge length of tetrahedra as the characteristic mesh size, and 2.878 when using the inverse cube root of the number of elements. The obtained values indicate good convergence of the implemented numerical scheme and confirm the correctness of the proposed approach.

Keywords: finite element method, convergence, elastic waves, anisotropic media
Citation in English: Smirnov I.E. Convergence study of the finite element method for elastic waves in anisotropic media // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1081-1091
Citation in English: Smirnov I.E. Convergence study of the finite element method for elastic waves in anisotropic media // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1081-1091
DOI: 10.20537/2076-7633-2026-18-5-1081-1091
URL: http://crm-en.ics.org.ru/journal/article/3771/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Smirnov I.E.

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