Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

Abstract. We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any poly-nomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincare ́ inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in L1. We derive optimal error estimates for any polyno-mial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have compa-rable size, which yields optimal anisotropic error estimates over n-rectangular domains. The interpolation theory extends to cases when the error and func-tion regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators. 1