Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

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• 作者：Ricardo H. Nochetto ; Enrique Otárola ; Abner J. Salgado
• 关键词：35J70 ; 35J75 ; 65D05 ; 65N30 ; 65N12
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：132
• 期：1
• 页码：85-130
• 全文大小：871 KB
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• 作者单位：Ricardo H. Nochetto (1)
  Enrique Otárola (2) (3)
  Abner J. Salgado (4)
  
  1. Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA
  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA
  3. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA
  4. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincaré 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 \(L^1\). We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over \(n\)-rectangular domains. The interpolation theory extends to cases when the error and function 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. Mathematics Subject Classification 35J70 35J75 65D05 65N30 65N12