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Nodal high-order methods on unstructured grids. I: Time-domain solution of Maxwell’s equations. (English) Zbl 1014.78016
The paper presents a high-order convergent method for the solution of the time-dependent Maxwell equations in geometrically complex domains. The spatial discretization on the unstructured tetrahedral mesh uses a high order nodal basis of multivariate Lagrange polynomials. The discrete equations are satisfied in a discontinuous Galerkin-Petrov form with the boundary and transmission conditions enforced weakly through a penalty term. The authors prove stability and convergence of the semidiscrete approximation to Maxwell’s equations and establish an error estimate of at most linear growth in time. The results can be applied to the solution of linear conservation laws. The authors present a number of numerical examples to verify the theoretical results and to illustrate the efficiency and robustness of the method when solving benchmark problems in computational electromagnetics. Problems of the efficient implementation and parallel performance are discussed.

78M25 Numerical methods in optics (MSC2010)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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