Optimal adaptivity for non-symmetric FEM/BEM coupling
This work provides a theoretical foundation for optimal adaptivity in non-symmetric coupled problems, addressing a key bottleneck in numerical analysis.
The authors prove general quasi-orthogonality for non-symmetric Johnson-Nedelec FEM/BEM coupling, enabling optimal convergence rates for adaptive algorithms. They establish that a standard adaptive algorithm achieves optimal rates, with techniques extendable to other problems like Stokes equation.
We develop a framework which allows us to prove the essential general quasi-orthogonality for the non-symmetric Johnson-Nedelec finite element/boundary element coupling. General quasi-orthogonality was first proposed in [Axioms of Adaptivity, 2014] as a necessary ingredient of optimality proofs and is the major difficulty on the way to prove rate optimal convergence of adaptive algorithms for many strongly non-symmetric problems. The proof exploits a new connection between the general quasi-orthogonality and LU-factorization of infinite matrices. We then derive that a standard adaptive algorithm for the Johnson-Nedelec coupling converges with optimal rates. The developed techniques are fairly general and can most likely be applied to other problems like Stokes equation.