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- Petr Vanek, Jan Mandel, Marian Brezina
- Computing
- 1996

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method… (More)

- Petr Vanek, Marian Brezina, Jan Mandel
- Numerische Mathematik
- 2001

We prove a convergence estimate for the Algebraic Multigrid Method with prolongations deened by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the… (More)

- Marian Brezina, Andrew J. Cleary, +5 authors John W. Ruge
- SIAM J. Scientific Computing
- 2001

We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new… (More)

Gauss-Seidel method is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers… (More)

- Jan Mandel, Marian Brezina
- Math. Comput.
- 1996

The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition… (More)

- Marian Brezina, Robert D. Falgout, Scott P. MacLachlan, Thomas A. Manteuffel, Stephen F. McCormick, John W. Ruge
- SIAM J. Scientific Computing
- 2004

- Jan Mandel, Marian Brezina, Petr Vanek
- Computing
- 1999

We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse basis functions. For a particular selection of the supports,… (More)

- Marian Brezina, Petr Vanek
- Computing
- 1999

We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence… (More)

- Marian Brezina, Tom Manteuffel, J . RUGE S . MCCORMICK
- 2008

Applying smoothed aggregation multigrid (SA) to solve a nonsymmetric linear system, Ax = b, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, √ AtA or… (More)

- James J. Brannick, Marian Brezina, +7 authors Ludmil Zikatanov
- 2005

1 Department of Applied Mathematics, University of Colorado at Boulder, Email: brannick@newton.colorado.edu, mbrezina@math.cudenver.edu, maclachl@colorado.edu, tmanteuf@colorado.edu, stevem@colorado.edu, jruge@colorado.edu 2 Department of Mathematical Sciences, Ball State University, Email: ilivshits@bsu.edu 3 Department of Applied Physics and Applied… (More)