Approximation and orthogonality in Sobolev spaces on a triangle

TL;DR

This paper investigates polynomial approximation in Sobolev spaces on a triangle, focusing on orthogonal structures and deriving sharp error estimates for derivatives up to second order.

Contribution

It explicitly constructs Sobolev orthogonal polynomials on a triangle and establishes sharp approximation error bounds for derivatives of order one and two.

Findings

Orthogonal polynomials in Sobolev spaces on a triangle are explicitly constructed.

Projection operators commute with partial derivatives in the Sobolev space.

Sharp error estimates are derived for polynomial approximation when r=1 and r=2.

Abstract

Approximation by polynomials on a triangle is studied in the Sobolev space $W_2^r$ that consists of functions whose derivatives of up to $r$-th order have bounded $L^2$ norm. The first part aims at understanding the orthogonal structure in the Sobolev space on the triangle, which requires explicit construction of an inner product that involves derivatives and its associated orthogonal polynomials, so that the projection operators of the corresponding Fourier orthogonal expansion commute with partial derivatives. The second part establishes the sharp estimate for the error of polynomial approximation in $W_2^r$, when $r = 1$ and $r=2$, where the polynomials of approximation are the partial sums of the Fourier expansions in orthogonal polynomials of the Sobolev space.