Core Concepts
The author presents an adaptive mesh refinement strategy based on T-coercivity to ensure quasi-optimality of the Galerkin finite element method for the Helmholtz equation.
Abstract
The content discusses an adaptive mesh refinement strategy based on T-coercivity to ensure quasi-optimality of the Galerkin finite element method for the Helmholtz equation. The approach focuses on criteria related to T-coercivity and weak T-coercivity, highlighting the dependence of mesh size on the gap between wave numbers and Laplace eigenvalues. The proposed adaptive scheme aims to generate optimal meshes with minimal degrees of freedom. Various theoretical results, numerical experiments, and applications on different geometries validate the effectiveness of the proposed strategy.
The content delves into detailed mathematical analyses, including definitions of T-coercivity and weak T-coercivity, well-posedness considerations for variational problems, and approximation using conforming Galerkin methods. It explores conditions ensuring quasi-optimality in discretizations through explicit bounds on mesh sizes relative to wave numbers and eigenvalues. The discussion extends to practical implementations with numerical experiments validating theoretical findings across different geometries.
Key points include discussions on coercivity issues in Helmholtz equation analysis, Schatz argument application, criteria based on T-coercivity for quasi-optimality assurance, and novel adaptive schemes for optimal mesh generation. Theoretical frameworks are supported by numerical validations showcasing convergence rates and solution quality under varying conditions.
Overall, the content provides a comprehensive exploration of adaptive mesh refinement strategies in mathematical contexts, emphasizing their significance in ensuring efficient solutions for complex equations like the Helmholtz equation.
Stats
k2 − λ(i∗)h = -4.82e+01 (N. DoFs: 409)
k2 − λ(i∗)h = -1.71e+01 (N. DoFs: 1469)
k2 − λ(i∗)h = -4.22e+00 (N. DoFs: 5545)
k2 − λ(i∗)h = -8.30e-01 (N. DoFs: 21521)
k2 − λ(i∗)h = -9.15e-03 (N. DoFs: 84769)
Quotes
"An adaptive scheme...aims to produce the mesh with minimal degrees of freedom."
"Theoretical results are validated through numerical experiments across various geometries."
"Adaptive strategies ensure quasi-optimal solutions while minimizing computational costs."