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# Institut für Mathematik - OPUS - Universität Augsburg

Institut für Mathematik - OPUS - Universität Augsburg

## to consider the

{v ∈ H 1 (Ω)|v| Σ = 0} and refer to H −1 Σ (Ω) as the associated dual space. For a simply connected polyhedral domain Ω with boundary Γ = ∂Ω we refer to H(curl; Ω) as the Hilbert space of vector fields q ∈ L 2 (Ω) such that ∇×q ∈ L 2 (Ω), equipped with the standard graph norm ‖ · ‖ curl,Ω . We denote by H 0 (curl; Ω) the subspace of vector fields with vanishing tangential trace components on Γ. We assume V and H to be Hilbert spaces of functions on Ω with inner products (·, ·) V , (·, ·) H and associated norms ‖ · ‖ V , ‖ · ‖ H such that V ⊂ H ⊂ V ∗ and V is continuously embedded in H. Given a bounded, V -elliptic bilinear form a(·, ·) : V × V → R and a bounded linear functional l ∈ V ∗ , we consider the variational equation: Find u ∈ V such that a(u, v) = l(v) , v ∈ V. (2.1) In view of the Lemma of Lax-Milgram [26], the variational equation (2.1) admits a unique solution u ∈ V . Example 1: A typical example is V = H0 1 (Ω), H = L 2 (Ω) and ∫ ( ) ∫ a(u, v) = a∇u · ∇v + cuv dx , l(v) := fv dx, (2.2) Ω where f ∈ L 2 (Ω), a = (a ij ) d i,j=1 , a ij ∈ L ∞ (Ω), 1 ≤ i, j ≤ d, is a symmetric, uniformly positive definite matrix-valued function, and c ∈ L ∞ + (Ω). Here, (2.1) represents the weak formulation of a second order elliptic boundary value problem. Example 2: Another example is V = H 0 (curl; Ω), H = L 2 (Ω) with ∫ ( ) ∫ a(u, v) = a(∇ × u) · (∇ × v) + cu · v dx , l(v) := f · v dx, (2.3) Ω where f ∈ L 2 (Ω), a ∈ L ∞ (Ω) such that a(x) ≥ a 0 > 0 a.e. in Ω, and c ∈ L ∞ (Ω). In case c ∈ L ∞ + (Ω), the variational equation (2.1) is the weak formulation of an H(curl)- elliptic boundary value problem, e.g., arising from a semi-discretization in time of the eddy currents equations. On the other hand, if c(x) < 0 a.e. in Ω as it is the case for the Helmholtz problem associated with the time-harmonic Maxwell equations, the bilinear form a(·, ·) is not V -elliptic, but satisfies a Gårding-type inequality. Under suitable assumptions on the data it can be shown that for the solution of (2.1) a Fredholm alternative holds true (cf., e.g., [45]). We assume (V i ) L i=0 , L ∈ N, V i = span{ϕ (i) 1 , · · · , ϕ(i) N i }, N i ∈ N, 0 ≤ i ≤ L, to be a nested sequence V i−1 ⊂ V i , 1 ≤ i ≤ L, of finite dimensional subspaces of V obtained, e.g., with respect to a nested hierarchy T i (Ω) of simplicial triangulations of Ω generated by the application of adaptive finite element methods to (2.1). For D ⊂ ¯Ω we refer to N i (D), E i (D), and F i (D) as the sets of nodes, edges, and faces of T i (Ω) in D. Moreover, for E ∈ E i (D), F ∈ F i (D), and T ∈ T i (Ω) we denote by h E the length of E, and by h F and h T the diameters of F and T , respectively. We set h i := max{h T | T ∈ T i (Ω)}. The Galerkin approximation of (2.1) with respect to V i ⊂ V, 0 ≤ i ≤ L, reads: Find u i ∈ V i such that a(u i , v i ) = l(v i ) , v i ∈ V i . (2.4) 3 Ω Ω

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