Polynomial finite element method for domains enclosed by piecewise conics

Abstract

We consider bivariate piecewise polynomial finite element spaces for curved domains bounded by piecewise conics satisfying homogeneous boundary conditions, construct stable local bases for them using Bernstein-Bézier techniques, prove error bounds and develop optimal assembly algorithms for the finite element system matrices. Numerical experiments confirm the effectiveness of the method.

Bivariate semialgebraic splines

“…The underlined postulation numbers in Table 1 behave roughly as 4r + 1, which is what is expected if the regularity of H 0 (J ) is 4r + 2. From the short exact sequence (7) we indeed expect reg(H 0 (J )) ≤ 4r + 2, since 4r + 2 is the regularity of the complete intersection S/ f r+1 , g r+1 . ⋄ Remark 6.3.…”

“…The last quotient is S/J(v) = S/J(v ′ ). From (7) we obtain (8) HP (H 0 (J ), d) = (2r + 2) 2 − HP(S/J(v), d) .…”

“…6.1] this fails for large r if two edge forms are tangent. In practice (see [7,8,9]) one may wish to impose vanishing conditions across a C 1 piecewise polynomial boundary, which requires such tangency at boundary vertices. Even in this situation it is likely possible to work out dimension formulas for small values of r using our approach.…”

“…A fundamental question is to determine the dimension of the vector space of splines on ∆ with a given smoothness and whose polynomial constituents have at most a fixed degree. Traditionally, ∆ is rectilinear, that is, a simplicial [23] or polygonal [21] complex, but it is useful to consider splines where the edges of ∆ are arcs of algebraic curves [7,8,9]. Such semialgebraic splines were studied by Wang and Chui [5,24,25], and by Stiller [22].…”

Approximation by $$C^1$$ C 1 Splines on Piecewise Conic Domains

“…In this section we estimate the error u − I △ u H k (Ω ) for functions u ∈ H m (Ω ) ∩ H 1 0 (Ω ), m = 5, 6. Similar to [7,Section 3], we follow the standard finite element techniques involving the Bramble-Hilbert Lemma (see [4,Chapter 4]) on the ordinary triangles, and make use of the estimate (3) on the pie-shaped triangles. Since the spline I △ u on the buffer triangles is constructed in part by interpolation and in part by the smoothness conditions, the estimate of the error on such triangles relies in particular on the estimates of the interpolation error on the neighboring ordinary and buffer triangles.…”