Sard kernel theorems on triangular domains with application to finite element error bounds

Abstract

Summary. Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.

Blending Function Interpolation: A Survey and Some New Results

“…(e.g., one less, a(h 3 ), for 2nd order problems). Apreeise statement of such error bounds, including the constants so that the bounds become computable, has been made by Barnhill and Gregor,y [12]. In this paper we are emphasizing the geometry and algebra of blending function methods, not their analysis.…”

Extension of Some Positive and Linear Operators to Domains with Curved Sides

“…Gordon, there have been constructed interpolation operators of Lagrange, Hermite and Birkhoff type, that interpolate the values of a given function or the values of the function and of certain of its derivatives on the boundary of a triangle with straight sides. These operators were applied in computer aided geometric design (see, e.g., [1]- [3], [5]]) and in finite element analysis (see, e.g., [1], [7], [8], [13], [17]- [19], [21], [22]). …”

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