By Cornelia Pester

His thesis is anxious with the finite aspect research and the a posteriori errors estimation for eigenvalue difficulties for normal operator pencils on two-dimensional manifolds. a selected program of the provided concept is the computation of nook singularities. Engineers use the data of the so-called singularity exponents to foretell the onset and the propagation of cracks. All result of this thesis are defined for 2 version difficulties, the Laplace and the linear elasticity challenge, and proven through various numerical effects.

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**Extra info for A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities**

**Example text**

16). 5 A lower error bound Proof. Note that F ([λh , uh ]) K×V˜ ≤ F ([λh , uh ]) K×V since V˜h ⊂ V . 22 imply that ηT2 1/2 F ([λh , uh ]) K×V˜h + F ([λh , uh ]) − F˜h ([λh , uh ]) K×V˜h F ([λh , uh ]) K×V + F ([λh , uh ]) − F˜h ([λh , uh ]) K×V˜h T ∈Th ≤ [λ0 , u0 ] − [λh , uh ] K×V ε2T + 1/2 .

Let x ∈ Nh and T ∈ ωx . The relation | v − π0,ωx v |0,T ≤ | v − π0,ωx v |0,ωx h+,T v 1,ωx holds for all v ∈ [H1 (ωx )]d . 14], where the result is called Friedrichs’ inequality. 14). 13 were proven in [6]. 13 was stated there in a more general form. 2), the above form is sufficient. The idea is to transform the given domain to a reference domain in R 2 and to apply appropriate estimates on the reference domain. To verify the Poincar´e inequality, a Bramble-Hilbert-type lemma is used, where averaged Taylor polynomials of v are employed, cf.

5 A lower error bound Proof. Note that F ([λh , uh ]) K×V˜ ≤ F ([λh , uh ]) K×V since V˜h ⊂ V . 22 imply that ηT2 1/2 F ([λh , uh ]) K×V˜h + F ([λh , uh ]) − F˜h ([λh , uh ]) K×V˜h F ([λh , uh ]) K×V + F ([λh , uh ]) − F˜h ([λh , uh ]) K×V˜h T ∈Th ≤ [λ0 , u0 ] − [λh , uh ] K×V ε2T + 1/2 .