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Extra resources for A Rational Finite Element Basis

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A central pnoblem in convergence analysis is to bound (from above) the error 11 Uh-u [IA by an expression of the form chS Ilull,, where c is a constant, the A- and Bnorms are meaningful in the sense that they give useful error measures, and s is as large as possible for the prescribed scheme. We demand that our wedge basis functions be regular (infinitely differentiable) within their associated polycons. Two properties of approximation by linear combination of these wedges play a significant role in convergence analysis: (1) order of continuity across polycon boundaries, and (2) degree of polynomial for which the wedges form a basis within each polycon.

The trapezoid exterior F i g . 4. T h e trapezoid exterior diagonal. diagonal is parallel to the parallel sides and passes through the intersection point of the other two sides. We note that the exterior diagonal is uniquely defined as the line that intersects the sides of the quadrilateral at all the exterior intersection points of these sides and at no other points. We have yet to establish property (5); the other 36 RATIONAL FINITE ELEMENT BASIS five properties in Section 1 . 5 are obviously satisfied.

5 WEDGE PROPERTIES. We now summarize properties thus far required of the wedge basis functions to achieve continuous patchwork degree one approximation over a collection of well-set polycons : (1) There is a node at each vertex and on each conic side. For each node there is an associated wedge within each polycon containing the node. ( 2 ) Wedge Wi(x,y) associated with node i is normalized to unity at node i. ( 3 ) Wedge Wi is linear on sides adjacent to i. Wedge Wi vanishes on sides opposite node i and (4) at all nodes j for which j # i.

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