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ROBUST MULTILEVEL METHODS FOR GENERAL SYMMETRIC POSITIVE DEFINITE OPERATORS ∗
"... Abstract. An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the pr ..."
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Abstract. An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of (nonlinear) algebraic multilevel iterations (AMLI). The crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems. The method is analyzed in a rather general setting and is applied to the scalar elliptic equation, the equations of linear elasticity, and equations arising in the solution of Maxwell’s equations. Numerical results for the scalar elliptic equation are presented showing its robust convergence behavior and large coarsening factors in the sequence of nested spaces.
ROBUST MULTILEVEL SOLVERS FOR HIGHCONTRAST ANISOTROPIC MULTISCALE PROBLEMS
"... Abstract. A robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented. The method, which belongs to the class of nonlinear algebraic multilevel iterations (AMLI), uses an abstract framework for general symmetric ..."
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Abstract. A robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented. The method, which belongs to the class of nonlinear algebraic multilevel iterations (AMLI), uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author. The crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems. A discussion on how to achieve large coarsening factors in this sequence of spaces, which is desirable in terms of computational complexity, is included. Particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal, i.e., do not constitute bases. Several numerical examples are provided verifying the theoretically established robustness results. 1.
Achieving robustness through coarse space enrichment in the two level Schwarz framework.
"... may suffer from a lack of robustness with respect to coefficient variation in the underlying set of PDEs. This is the case in particular if the partition into subdomains is not aligned with all jumps in the coefficients. Thanks to the theoretical analysis of two level Schwarz methods (see [11] and r ..."
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may suffer from a lack of robustness with respect to coefficient variation in the underlying set of PDEs. This is the case in particular if the partition into subdomains is not aligned with all jumps in the coefficients. Thanks to the theoretical analysis of two level Schwarz methods (see [11] and references therein) this lack of robustness can be traced back to the so called stable splitting property (already in [4]). Following the same ideas as in the pioneering work [1] we propose to solve a generalized eigenvalue problem in each subdomain which identifies which vectors are responsible for slow convergence. The spectral problem is specifically chosen to separate components that violate the stable splitting property. These vectors are then used to span the coarse space which is taken care of by a direct solve while all remaining components can be resolved on the subdomains. The result is a preconditioned system with a condition number estimate that does not depend on the number of subdomains or any jumps in the coefficients. We refer to this method as GenEO for Generalized Eigenproblems in the Overlaps. It is closely related to the work of [2] where the same strategy leads to a different eigenproblem and different condition
SPECTRAL COARSE SPACES IN ROBUST TWOLEVEL METHODS
"... Abstract. A survey of recently proposed approaches for the construction of spectral coarse spaces is provided. These coarse spaces are in particular used in twolevel preconditioners. At the core of their construction are local generalized eigenvalue problems. It is shown that by means of employing ..."
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Abstract. A survey of recently proposed approaches for the construction of spectral coarse spaces is provided. These coarse spaces are in particular used in twolevel preconditioners. At the core of their construction are local generalized eigenvalue problems. It is shown that by means of employing these spectral coarse spaces in twolevel additive Schwarz preconditioners one obtains preconditioned systems whose condition numbers are independent of the problem sizes and problem parameters such as (highly) varying coefficients. A unifying analysis of the recently presented approaches is given, pointing out similarities and differences. Some numerical experiments confirm the analytically obtained robustness results. 1.