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Numerical treatment of PDEs. Finite element method by Hiptmair R., Schwab C.

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By Hiptmair R., Schwab C.

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Thanks to the density theorem Thm. 52 we have the equivalent definition u H −1 (Ω) := sup v∈C0∞ (Ω)\{0} Ω uv dξ v H 1 (Ω) . It can be shown that this norm arises from some inner product on L2 (Ω). 57. 35 is called H −1 (Ω). 58. The space H −1 (Ω) is a Hilbert space, which is isometrically isomorphic ∗ to (H01 (Ω)) . Proof. For any u ∈ L2 (Ω) the mapping fu : v ∈ H01 (Ω) → functional on H01 (Ω) with norm fu ∗ (H01 (Ω)) fu , v = sup ∗ (H01 (Ω)) ×H01 (Ω) v v∈H01 (Ω) = sup v∈H01 (Ω) H 1 (Ω) Ω uv dξ v H 1 (Ω) Ω uv dξ is a continuous = u H −1 (Ω) .

D ) ∈ Nn0 . Set |α| := α1 + · · · + αn and denote by ✞ ∂ := ∂ α1 ∂ αn · · · ∂ξ1α1 ∂ξnαn the partial derivative of order |α|. Remember that for sufficiently smooth functions all partial derivatives commute. Provided that the derivatives exist, the gradient of a function f : Ω ⊂ Rd → R is the column vector grad f (ξ) := ∂f ∂f (ξ), . . , (ξ) ∂ξ1 ∂ξ1 T , ξ∈Ω. The divergence of a vector field f = (f1 , . . , fd ) : Ω ⊂ Rd → Rd is the function d div f (ξ) := k=1 ∂fk (ξ) , ∂ξk ξ∈Ω. The differential operator ∆ := div ◦ grad is known as Laplacian.

13. Assume that A and c are bounded and continuously differentiable on Ω. If u ∈ C 1 (Ω) ∩ C 0 (Ω), j ∈ (C 1 (Ω))d ∩ (C 0 (Ω))d satisfy (FL), (EL) in a pointwise sense, and the prescribed boundary conditions, then these functions are called a classical solution of the boundary value problem. 14. In general it is impossible to establish existence and uniqueness of classical solutions. This can be achieved for constant coefficients and pure Dirichlet or Neumann boundary conditions, see [13, Vol. I].

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