Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MIXED FINITE VOLUME METHOD FOR TWO-DIMENSIONAL MAXWELL'S EQUATIONS

  • Kwang-Yeon Kim (Department of Mathematics Kangwon National University) ;
  • Do Young Kwak (Department of Mathematical Sciences Korea Advanced Institute of Science and Technology)
  • Received : 2023.11.06
  • Accepted : 2024.05.27
  • Published : 2025.01.01
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Abstract

We propose and analyze a mixed finite volume method for the two-dimensional time-harmonic Maxwell's equations which simultaneously approximates the vector field 𝒖 and the scalar function ξ = µ-1 curl 𝒖. The method chooses the lowest-order Nédélec edge element for u and the P1 Crouzeix-Raviart nonconforming element for ξ on triangular meshes. It is shown that the method is reduced to a modified P1 nonconforming FEM for ξ or a modified edge element method for 𝒖 by eliminating the discrete variable of 𝒖 or ξ. After solving the reduced method, the eliminated discrete variable can be recovered from the other one via a simple local formula. Using this feature, we also derive optimal a priori error estimates under weak regularity assumptions and show that the approximation to ξ has a higher-order of convergence in the L2 norm than the one obtained by direct differentiation of the approximation to 𝒖 when the exact solution is sufficiently smooth.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1F1A1050243).

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