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BDDC
In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of coarse degrees of freedom, which can be values at the corners of the subdomains, or averages over the edges or the faces of the interface between the subdomains. One application of the BDDC preconditioner then combines the solution of local problems on each subdomains with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. With a proper choice of the coarse degrees of freedom (corners in 2D, corners plus edges or corners plus faces in 3D) and with regular subdomai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FETI-DP
The FETI-DP method is a domain decomposition methodC. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, ''FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method'', Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523--1544. that enforces equality of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur.J. Mandel and R. Tezaur, ''On the convergence of a dual-primal substructuring method'', Numerische Mathematik, 88 (2001), pp. 543--558. The method was further improved by enforcing the equality of averages across the edges or faces on subdomain interfacesC. Farhat, M. Lesoinne, and K. Pierson, ''A scalable dual-primal domain decomposition method'', Numer. Linear Algebra Appl., 7 (2000), pp. 687--714. Preconditioning techniques for large sparse matrix problems in industrial applications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Mandel
Jan Mandel is a Czech-American mathematician. He received his PhD from the Faculty of Mathematics and Physics, Charles University in Prague and was a Senior Research Scientist there. Since 1986, he is professor of Mathematics at the University of Colorado Denver. Since 2013, he is senior scientist at the Institute of Computer Science of the Czech Academy of Sciences. He has worked in the field of multigrid methods and domain decomposition methods. He developed the balancing domain decomposition method and, with coauthors, published the convergence proofs of the FETI, FETI-DP, and BDDC methods, and the proof of the equivalence of the FETI-DP and the BDDC methods. He has been involved in the field of dynamic data driven application systems and data assimilation with applications in wildfire and epidemic modeling. He has contributed to the WRF-Fire WRF-SFIRE is a coupled Atmospheric model, atmosphere-Wildfire modeling, wildfire model, which combines the Weather Research and Forecas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balancing Domain Decomposition
In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method.J. Mandel, ''Balancing domain decomposition'', Comm. Numer. Methods Engrg., 9 (1993), pp. 233–241. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain nullspaces. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in oil reservoir simulation by mixed finite elements.L. C. Cowsar, J. Mandel, and M. F. Wheeler, ''Balancing domain decomposition for mixed finite elements'', Math. Comp., 64 (1995), pp. 989–1015. In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D. For 4th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coarse Problem
: ''This article deals with a component of numerical methods. For coarse space in topology, see coarse structure.'' In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower resolution, retaining its essential characteristics, but with fewer variables. The purpose of the coarse problem is to propagate information throughout the whole problem globally. In multigrid methods for partial differential equations, the coarse problem is typically obtained as a discretization of the same equation on a coarser grid (usually, in finite difference methods) or by a Galerkin approximation on a subspace, called a coarse space. In finite element methods, the Galerkin approximation is typically used, with the coarse space generated by larger elements on the same domain. Typically, the coarse problem corresponds to a grid that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Decomposition Method
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG. In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. Mathematical formulation Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two.Boresi, A. P. and Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced mechanics of materials, John Wiley and Sons, New York. When the length is considerably longer than the width and the thickness, the element is called a beam. For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Susanne Brenner
Susanne Cecelia Brenner is an American mathematician, whose research concerns the finite element method and related techniques for the numerical solution of differential equations. She is a Boyd Professor at Louisiana State University. Previously, she held the Nicholson Professorship of Mathematics and the Michael F. and Roberta Nesbit McDonald Professorship at Louisiana State University, She currently chairs the editorial committee of the journal ''Mathematics of Computation''. During 2021-2022 she is serving as President of the Society for Industrial and Applied Mathematics (SIAM). Education and career Brenner did her undergraduate studies in mathematics and German at West Chester State College and received a master's degree in mathematics from SUNY Stony Brook.Curriculum vitae Retrieved 2013-10-15. She obtained her |
Olof B
Olov (or Olof) is a Swedish form of Olav/Olaf, meaning "ancestor's descendant". A common short form of the name is ''Olle''. The name may refer to: *Per-Olov Ahrén (1926–2004), Swedish clergyman, bishop of Lund from 1980 to 1992 *Per-Olov Brasar (born 1950), retired professional ice hockey forward *Olov Englund (born 1983), Swedish bandy player *Per Olov Enquist (1934–2020), one of Sweden's internationally best known authors * Olle Hagnell (1924–2011), Swedish psychiatrist *Karl Olov Hedberg (1923–2007), botanist, taxonomist, author, professor at Uppsala University *Olle Hellbom (1925–1982), Swedish film director *Per Olov Jansson (1920–2019), Finnish photographer *Olof Johansson (born 1937), Swedish politician *Per-Olov Kindgren (born 1956), Swedish musician, composer, guitarist and music teacher *Olov Lambatunga, Archbishop of Uppsala, Sweden, 1198–1206 *Sven-Olov Lawesson (1926–1988), Swedish chemist known for his popularization of Lawesson's reagent within the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. In linear regression the condition number of the moment matrix can be used as a diagnostic for multicollinearity. The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
