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A posteriori error estimates for the virtual element method.

Andrea Cangiani1, Emmanuil H Georgoulis1,2, Tristan Pryer3

  • 1Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH UK.

Numerische Mathematik
|November 21, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces a computable residual-type error estimator for the virtual element method (VEM) in elliptic problems. This method simplifies adaptive mesh refinement on complex meshes without post-processing.

Keywords:
65G9965N1265N30

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • The virtual element method (VEM) offers a flexible approach for solving elliptic problems on complex meshes.
  • Posteriori error estimation is crucial for guiding adaptive mesh refinement in numerical methods.
  • Existing methods may face challenges with complex mesh geometries and require post-processing.

Purpose of the Study:

  • To develop and analyze a residual-type posteriori error estimator for VEM applied to general elliptic problems.
  • To demonstrate the computability and reliability of the proposed error estimator.
  • To leverage the estimator for efficient and simplified adaptive mesh refinement.

Main Methods:

  • Developed a residual-type error estimator based on VEM solution's degrees of freedom and element-wise polynomial projection.
  • Proved upper and lower bounds for the error estimator relative to the VEM approximation error.
  • Implemented adaptive mesh refinement driven by the proposed error estimator on various test problems.

Main Results:

  • The error estimator is fully computable using only VEM solution data.
  • The estimator provides proven upper and lower bounds on the approximation error.
  • Adaptive mesh refinement is simplified, allowing for complex mesh elements without post-processing.

Conclusions:

  • The presented residual-type error estimator is effective and efficient for VEM in elliptic problems.
  • The method significantly simplifies adaptive mesh refinement, even for complex polygonal/polyhedral meshes.
  • This work contributes to the robustness and practical applicability of the virtual element method.