Finite Element Analysis. Barna Szabó

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On plotting vs. N on log‐log scale a straight line with the slope will be seen for sufficiently large N. The estimated value of β, corresponding to the ith solution in the sequence, is denoted by βi. It is computed from eq. (1.98):

      (1.102)

      Examples

      The properties of the finite element solution with reference to a family of model problems is discussed in the following. The problems are stated as follows: Find such that

(1.103)

      where κ and c are constants and is defined such that the exact solution is:

(1.104)

      As explained in Section 1.5.1, when α is not an integer, the case considered in the following, then this solution lies in the space . Therefore the asymptotic rate of h‐convergence on uniform meshes, predicted by eq. (1.92), is and the asymptotic rate of p‐convergence on a fixed mesh is .

      We selected this problem because it is representative of the singular part of the exact solutions of two‐and three‐dimensional elliptic boundary value problems.

      Referring to Theorem 1.3, we have for all therefore . Consequently for the kth element the load vector in the local numbering convention is:

(1.105)

      where by definition .

When then the first derivative of is infinity in the point . To avoid having in the integrand, the first term in eq. (1.105) is integrated by parts:

      Since for and , we have:

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