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Additional resources for A rational finite element basis, Volume 114 (Mathematics in Science and Engineering)
5. Quadrilateral wedge W 1' A m A L COORDINATES AS LIMITS OF RATIONAL WEDGES As one of the interior angles of a quadrilateral is increased to IT,the four rational wedges approach functions, only one of which in this ill-set limit 37 THE QUADRILATERAL is continuous. Two linear combinations of the other three discontinuous limit functions can be found which are continuous. The three continuous functions thereby obtained are the triangle basis functions (that is, the areal coordinates for the triangle).
It has been shown(Strang and Fix, 1973) that for a wide class of problems a finite element space of degree k - 1 over each element achieves approximation of order hk to an arbitrary smooth function and of order hk-s to its derivatives of order s. 16b) i=l i=l We require that our wedges satisfy these equations. 6) is linear and is uniquely determined Eqs by the three non-collinear vertex values. 16) are obviously satisfied for the triangle. Let ui be the value of linear function u at vertex i of a parallelogram.
Similarly, It is thus shown that the discontinuous limit functions of the quadrilateral wedges may be combined to yield the continuous linear basis functions for the limiting triangle. Areal coordinates are a degenerate form of rational quadrilateral wedges. 4 AN EXAMPLE OF QUADRILATERAL WEDGES By way of illustration, we determine the wedges for a sample quadrilateral. Referring to Fig. 7, we have (4;l) = y, (1;2) = (2y - 3 x ) / m , - 8 y ) / m , (3;4) = (4 - 2~ (2;3) = (5 + 2~ Q1 = (20 + 8~ 17y)/m, - - y)/&‘, and the rational basis functions for degree one approximation over the quadrilateral are: 39 THE QUADRILATERAL Fig.