New PDF release: 33 years of numerical instability

By Dahlquist G.

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Example text

In this section, we will review some results of the finite element method. There are some basic aspects in the construction of finite element approximations. First we need a partition (or triangulation) of the domain of the differential equation into sub-domains called elements. 7. A finite element mesh define a finite element space. Finally, we choose basis functions for the finite element space. The basis functions should have small supports. For simplicity, we assume R is a two-dimensional polygonal domain.

Convergence of the method can also be achieved by using piecewise increasingly higher degree polynomials over relatively coarse finite element meshes, leading to the p-version finite element method. Detailed discussion of the p-version finite element method can be found in [146]. The p-version approximation is more efficient in areas where the solution is smooth, so it is natural to combine the ideas of the p-version and the h-version to make the finite element method very efficient on many problems.

If ai is not a node of K , then $ilK = 0. 30 We examine an example of linear elements. Assume R c IR2 is a poly onal domain, which is triangulated into triangles K , K E P h . e. e. the vertices of the lie in 12; and {ai}2Nint+l triangulation that lie on 8 0 . From each vertex ai, construct K~ as the patch of the elements K which contain ai as a vertex. ~~ associated with the node ai is a continuous function in 2,which is linear on each K and . corresponding piecewise linear function space is is non-zero only in K ~ The then X h = span{41i, 1 5 i 5 Nh}.

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33 years of numerical instability by Dahlquist G.


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