Time-Domain Finite Element Methods for Metamaterials
In this chapter, we present several fully discrete mixed finite element methods for solving Maxwell’s equations in metamaterials described by the Drude model and the Lorentz model. In Sects. 3.1 and 3.2, we respectively discuss the constructions of diverg
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Time-Domain Finite Element Methods for Metamaterials
In this chapter, we present several fully discrete mixed finite element methods for solving Maxwell’s equations in metamaterials described by the Drude model and the Lorentz model. In Sects. 3.1 and 3.2, we respectively discuss the constructions of divergence and curl conforming finite elements, and the corresponding interpolation error estimates. These two sections are quite important, since we will use both the divergence and curl conforming finite elements for solving Maxwell’s equations in the rest of the book. The material for Sects. 3.1 and 3.2 is quite classic, and we mainly follow the book by Monk (Finite element methods for Maxwell’s equations. Oxford Science Publications, New York, 2003). After introducing the basic theory of divergence and curl conforming finite elements, we focus our discussion on developing some finite element methods for solving the time-dependent Maxwell’s equations when metamaterials are involved. More specifically, in Sect. 3.3, we discuss the well posedness of the Drude model. Then in Sects. 3.4 and 3.5, we present detailed stability and error analysis for the Crank-Nicolson scheme and the leap-frog scheme, respectively. Finally, we extend our discussion on the well posedness, scheme development and analysis to the Lorentz model and the DrudeLorentz model in Sects. 3.6 and 3.7, respectively.
3.1 Divergence Conforming Elements 3.1.1 Finite Element on Hexahedra and Rectangles If a vector function has a continuous normal derivative, then such a finite element is usually called divergence conforming. More specifically, similar to the H 1 conforming finite elements discussed in Chap. 2, we can prove the following result.
J. Li and Y. Huang, Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Series in Computational Mathematics 43, DOI 10.1007/978-3-642-33789-5 3, © Springer-Verlag Berlin Heidelberg 2013
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3 Time-Domain Finite Element Methods for Metamaterials
Lemma 3.1. Let K1 and K2 be two non-overlapping Lipschitz domains having a common interface such that K1 \ K2 D : Assume that u1 2 H.divI K1 / and u2 2 H.divI K2 /, and u 2 .L2 .K1 [ K2 [ //d be defined by uD
u1 u2
on K1 ; on K2 :
Then u1 n D u2 n on implies that u 2 H.divI K1 [ K2 [ /, where n is the unit normal vector to . Proof. Suppose that we have a function u 2 .L2 .K1 [ K2 [ //d defined by ujKi D ui ; i D 1; 2; and u1 n D u2 n on . To prove that u 2 H.divI K1 [ K2 [ /, we only need to show that r u 2 L2 .K1 [ K2 [ /. For any function 2 C01 .K1 [ K2 [ /, using integration by parts, we have Z u rd x K1 [K2 [
Z
D
Z
Z
r .ujK1 /d x K1
r .ujK2 /d x C K2
.u1 n1 C u2 n2 /ds;
where n1 and n2 denote the unit outward normals to @K1 and @K2 , respectively. Denote a function v such that vjKl D r .ujKl /; l D 1; 2. Using the assumption that u1 n D u2 n on , we see that the boundary integral term vanishes. Hence, we have Z Z u rd x D vd x; K1 [K2 [
K1 [K2 [
which show
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