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An optimal adaptive wavelet method for first order system least squares.

Nikolaos Rekatsinas1, Rob Stevenson1

  • 1Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands.

Numerische Mathematik
|August 14, 2018
PubMed
Summary

This study demonstrates that any well-posed second-order partial differential equation (PDE) can be transformed into a first-order least squares system. This system is then efficiently solved using an adaptive wavelet method for optimal computational performance.

Keywords:
41A2542C4047J2565J1565N1265N3065T6076D05

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Second-order partial differential equations (PDEs) are fundamental in modeling various scientific phenomena.
  • Solving complex PDEs often requires sophisticated numerical techniques.
  • Existing methods may face challenges with efficiency or specific boundary conditions.

Purpose of the Study:

  • To present a novel reformulation of any well-posed second-order PDE into a first-order least squares system.
  • To introduce an adaptive wavelet solver for this reformulated system.
  • To analyze the computational complexity and applicability of the proposed method.

Main Methods:

  • Reformulation of second-order PDEs into first-order least squares systems.
  • Development and application of an adaptive wavelet solver.
  • Analysis of computational complexity and convergence properties.

Main Results:

  • Demonstration that any well-posed second-order PDE can be converted into a well-posed first-order least squares system.
  • The adaptive wavelet solver achieves optimal computational complexity.
  • Successful application to second-order elliptic PDEs with inhomogeneous boundary conditions and stationary Navier-Stokes equations.

Conclusions:

  • The proposed reformulation and adaptive wavelet solver offer an efficient and general approach for solving second-order PDEs.
  • This method provides a unified framework for a range of challenging PDE problems.
  • The findings have significant implications for computational science and engineering.