A posteriori error estimates

Definition

A posteriori error estimates are methods used to assess the accuracy of a numerical solution after the computation has been completed. These estimates provide a way to quantify the difference between the computed solution and the exact solution, helping to identify how reliable the results are. By analyzing the error, adjustments can be made to improve the solution or refine the computational approach.

5 Must Know Facts For Your Next Test

1. A posteriori error estimates are often computed using residuals from the numerical method, which measure how much the computed solution fails to satisfy the original problem's equations.
2. These estimates can guide adaptive methods, allowing for dynamic refinement of meshes or time steps in response to estimated errors.
3. They are particularly useful in finite element methods, where they can help determine where more computational resources should be allocated.
4. A posteriori error estimates can also be used to establish convergence rates for numerical methods, providing insights into how quickly a solution is approaching the exact answer.
5. The accuracy of a posteriori error estimates depends on the choice of norms and can vary significantly based on the problem being solved.

Review Questions

• How do a posteriori error estimates contribute to improving numerical solutions?
  • A posteriori error estimates help improve numerical solutions by quantifying the difference between computed and exact solutions after calculations are completed. By analyzing these errors, one can identify regions where accuracy is lacking and implement mesh refinement or adaptivity strategies. This targeted improvement ensures that computational resources are focused on areas that most influence overall solution quality.
• What role do residuals play in calculating a posteriori error estimates, and how can they impact subsequent computations?
  • Residuals serve as a critical component in calculating a posteriori error estimates as they measure how well the computed solution satisfies the governing equations. The larger the residual, the greater the estimated error, indicating that adjustments may be necessary. By using residuals to guide adaptive methods or refine meshes, subsequent computations can achieve higher accuracy where it matters most, ultimately leading to more reliable results.
• Evaluate the importance of selecting appropriate norms when using a posteriori error estimates in numerical analysis.
  • Selecting appropriate norms is crucial when using a posteriori error estimates because different norms can yield vastly different assessments of error magnitude. The choice of norm affects how errors are quantified and understood, which in turn influences decisions made during adaptive refinement or convergence analysis. Understanding this relationship allows for better-informed choices in numerical methods, ensuring that results are not only accurate but also meaningful within the context of specific problems being addressed.

Related terms

Convergence: The property of a numerical method where the computed solution approaches the exact solution as the step size or discretization parameter is reduced.

Mesh Refinement: A technique in numerical analysis that involves creating a finer grid or mesh in regions where higher accuracy is needed to improve solution accuracy.

Error Analysis: The study of the types and sources of errors in numerical computations, aiming to understand their impact on the final results.