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High technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these words - nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which

Social sciences --- Qualitative research --- Qualitative analysis (Research) --- Qualitative methods (Research) --- Research --- Data processing. --- Coupled problems (Complex systems) --- Mathematical optimization --- Nonlinear theories --- 519.61 --- 681.3*G4 --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- 681.3*G4 Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- 519.61 Numerical methods of algebra --- Numerical methods of algebra --- Coupled field problems (Complex systems) --- Problems, Coupled (Complex systems) --- Dynamics

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During the last years, scientific computing has become an important research branch located between applied mathematics and applied sciences and engineering. Highly efficient numerical methods are based on adaptive methods, higher order discretizations, fast linear and non-linear iterative solvers, multi-level algorithms, etc. Such methods are integrated in the adaptive finite element software ALBERTA. It is a toolbox for the fast and flexible implementation of efficient software for real life applications, based on modern algorithms. ALBERTA also serves as an environment for improving existent, or developing new numerical methods in an interplay with mathematical analysis and it allows the direct integration of such new or improved methods in existing simulation software.

Computer algorithms. --- Computer software --- Finite element method --- Science --- Development. --- Computer programs. --- Data processing. --- Computer algorithms --- Data processing --- Computer programs --- Development --- Computer science. --- Computer software. --- Software engineering. --- Computational Science and Engineering. --- Mathematical Software. --- Mathematics of Computing. --- Software Engineering/Programming and Operating Systems. --- Computer software engineering --- Engineering --- Software, Computer --- Computer systems --- Informatics --- 519.63 --- 681.3 *G18 --- 681.3*G4 --- 681.3*G4 Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- Algorithms --- Development of computer software --- Software development --- Electronic data processing --- Computer mathematics. --- Computer science—Mathematics. --- Computer mathematics --- Mathematics --- Science - Data processing --- Finite element method - Computer programs --- Computer software - Development --- Acqui 2006