Goal-oriented adaptive finite element methods with optimal computational complexity.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Partial Differential Equations

Background:

• Linear symmetric and elliptic partial differential equations (PDEs) are fundamental in various scientific and engineering domains.
• Efficiently solving these PDEs, especially with complex goal functionals, requires advanced numerical methods.
• Existing adaptive finite element methods often focus on error reduction concerning degrees of freedom, not overall computational cost.

Purpose of the Study:

• To design and analyze a novel goal-oriented adaptive finite element method (GOAFEM).
• To steer adaptive mesh-refinement and the solution of linear systems using efficient iterative solvers.
• To prove optimal convergence rates with respect to total computational cost.

Main Methods:

• Design of a GOAFEM tailored for linear symmetric and elliptic PDEs with linear goal functionals.
• Integration of contractive iterative solvers, such as optimally preconditioned conjugate gradient or geometric multigrid, for linear system approximation.
• Theoretical analysis to establish convergence properties of the adaptive algorithm.

Main Results:

• Demonstrated linear convergence of the proposed adaptive algorithm.
• Achieved optimal algebraic convergence rates.
• Proved optimal complexity, signifying convergence rates relative to total computational cost, a key advancement over prior work.

Conclusions:

• The developed GOAFEM provides an efficient approach for solving linear PDEs with goal-oriented adaptivity.
• The method achieves optimal computational complexity, making it highly efficient for practical applications.
• This work advances the state-of-the-art in adaptive finite element methods by considering total computational cost.