@inproceedings {42935,
	title = {Rational approximation preconditioners for multiphysics problems},
	journal = {Numerical Methods and Applications (NMA 2022)},
	volume = {13858},
	year = {2023},
	month = {05/2023},
	pages = {100-113},
	publisher = { Springer},
	edition = {Lecture Notes in Computer Science},
	address = {Cham},
	abstract = {We consider a class of mathematical models describing multiphysics phenomena interacting through interfaces. On such interfaces, the traces of the fields lie (approximately) in the range of a weighted sum of two fractional differential operators. We use a rational function approximation to precondition such operators. We first demonstrate the robustness of the approximation for ordinary functions given by weighted sums of fractional exponents. Additionally, we present more realistic examples utilizing the proposed preconditioning techniques in interface coupling between Darcy and Stokes equations.},
	keywords = {Multiphysics, preconditioning, Rational approximation},
	isbn = {978-3-031-32412-3},
	doi = {10.1007/978-3-031-32412-3_9},
	url = {https://doi.org/10.1007/978-3-031-32412-3_9},
	author = {Budisa, Ana and Hu, Xiaozhe and Kuchta, Miroslav and Mardal, Kent-Andre and Ludmil Tomov {Zikatanov}},
	editor = {Georgiev, Ivan and Datcheva, Maria and Georgiev, Krassimir and Nikolov, Geno}
}
@article {doi:10.1002/num.22304,
	title = {Preconditioning trace coupled 3d-1d systems using fractional Laplacian},
	journal = {Numerical Methods for Partial Differential Equations},
	volume = {35},
	number = {1},
	pages = {375-393},
	publisher = { Wiley},
	abstract = {Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work, we consider a simplified model problem of a 3d-1d coupling and the main objective is to construct algorithms that may utilize standard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three-dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well-defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.},
	keywords = {Lagrange multipliers, preconditioning, saddle-point problem, trace},
	doi = {10.1002/num.22304},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/num.22304},
	author = {Kuchta, Miroslav and Mardal, Kent-Andre and Mortensen, Mikael}
}
@article {doi:10.1002/nla.2212,
	title = {On the singular Neumann problem in linear elasticity},
	journal = {Numerical Linear Algebra with Applications},
	volume = {26},
	number = {1},
	note = {e2212 nla.2212},
	pages = {e2212},
	publisher = { Wiley},
	abstract = {Summary The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a nonsingular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation, we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lam{\'e} constant are constructed for the pure displacement formulations, whereas a preconditioner that is robust in both Lam{\'e} constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. On the basis of this observation, a modification to the conjugate gradient method is proposed, which achieves optimal error convergence of the computed solution.},
	keywords = {conjugate gradient, Linear elasticity, preconditioning, rigid motions, singular problems},
	doi = {10.1002/nla.2212},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2212},
	author = {Kuchta, Miroslav and Mardal, Kent-Andre and Mortensen, Mikael}
}
