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Data assimilation for the heat equation using stabilized finite element methods.

Erik Burman1, Lauri Oksanen1

  • 1Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.

Numerische Mathematik
|July 6, 2018
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Summary
This summary is machine-generated.

This study presents a novel data assimilation method for the heat equation using stabilized finite elements. The approach ensures a unique solution and provides accurate error estimates for improved model stability.

Keywords:
65M1265M1565M3065M32

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Data assimilation is crucial for solving inverse problems governed by partial differential equations (PDEs).
  • The heat equation, a fundamental PDE, often requires robust numerical methods for accurate solutions in data assimilation contexts.
  • Ill-posedness in continuous PDE models necessitates regularization techniques for stable numerical solutions.

Purpose of the Study:

  • To develop and analyze an optimization-based data assimilation framework for the heat equation.
  • To incorporate stabilized finite element methods to enhance the stability and accuracy of the numerical scheme.
  • To derive rigorous error estimates that account for both the finite element approximation and the continuous model's stability properties.

Main Methods:

  • Finite element space semi-discretization of the heat equation.
  • Optimization-based approach for data assimilation.
  • Application of stabilized finite element theory for regularization operator design.
  • Derivation of error estimates using numerical stability and conditional stability estimates.

Main Results:

  • The space semi-discretized system is proven to admit a unique solution.
  • Error estimates are derived, reflecting the approximation order of the finite element space and the continuous model's stability.
  • The framework is illustrated with two distinct data assimilation scenarios exhibiting different stability characteristics.

Conclusions:

  • The proposed data assimilation method, leveraging stabilized finite elements, provides a robust framework for the heat equation.
  • The derived error estimates offer valuable insights into the interplay between numerical discretization and continuous model stability.
  • The methodology is adaptable and applicable to various data assimilation problems with differing stability properties.