The approximation of the Maxwell eigenvalue problem using a leastsquares method
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In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two divcurl systems with complementary boundary conditions. We formulate each of these divcurl systems as a general variational problem with different test and trial spaces, i.e., the solution space is L 2(Ω) ≡ (L 2(Ω)) 3 and components in the test spaces are in subspaces of H 1(Ω), the Sobolev space of order one on the computational domain Ω. A finiteelement leastsquares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the divcurl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given. © 2005 American Mathematical Society.
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Bramble, J. H., Kolev, T. V., & Pasciak, J. E.
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Divcurl Systems

Finite Element Approximation

Infsup Condition

Maxwell Eigenvalues

Maxwell's Equations

Negative Norm Leastsquares
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