By Cornelia Pester

ISBN-10: 3832512497

ISBN-13: 9783832512491

His thesis is anxious with the finite point research and the a posteriori blunders estimation for eigenvalue difficulties for normal operator pencils on two-dimensional manifolds. a selected program of the offered conception 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 tested by way of a number of numerical effects.

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

**Example text**

4. 8) for all T ∈ Th with ϑ−,T > 0. The following lemma is a simple consequence of this inequality. 6. 9) holds for all elements T ∈ Th with ϑ−,T > 0. Proof. 3]. For completeness, the proof is repeated. It is clear that ϑ+,T ≥ ϑ−,T . It remains to show ϑ+,T ϑ−,T . By definition, there are two angles θ− and θ+ with ϑ−,T = sin θ− and ϑ+,T = sin θ+ . Obviously, |θ+ − θ− | ≤ hSθ,T . 8) and sin x ≤ |x| ∀x ∈ R imply that sin(θ+ − θ− ) ≤ |θ+ − θ− | ≤ hSθ,T ϑ−,T , and therefore ϑ+,T = sin θ+ = sin θ− cos(θ+ − θ− ) + cos θ− sin(θ+ − θ− ) = ϑ−,T cos(θ+ − θ− ) + ϑ−1 −,T cos θ− sin(θ+ − θ− ) ϑ−,T .

If there are constants R0 > 0, c0 > 0 such that B (λ1 )u1 − B (λ0 )u0 V + B(λ1 ) − B(λ0 ) for all [λ1 , u1 ] ∈ K × V with [λ1 , u1 ] − [λ0 , u0 ] continuous at [λ0 , u0 ] with some constant γ. 26) ≤ R0 , then the operator DF is Lipschitz Proof. We have to show that there is a constant γ > 0 such that DF ([λ1 , u1 ]) − DF ([λ0 , u0 ]) L(K×V,K×V ) sup sup [λ, u] ∈ K × V [λ, u] K×V = 1 [µ, v] ∈ K × V [µ, v] K×V = 1 = ≤ γ [λ1 , u1 ] − [λ0 , u0 ] K×V DF ([λ1 , u1 ])([λ, u]) − DF ([λ0 , u0 ])([λ, u]), [¯ µ, v¯] .

When we speak about vector functions, we mean first order tensor-valued functions, unless a basis in Rd is specified. Usually, tensors are developed n times into the same basis, that is, b1i = b2i = . . = bni , i = 1, 2, 3. This is useful to simplify the further calculations, when only co- and contravariant tensor bases are considered [50, 78], and makes sense in many cases, for example, to determine the tensor invariants or the components of a material tensor. This restriction is less convenient, however, when we want to define the gradient of a vector function u on the sphere.

### A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities by Cornelia Pester

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