Numerical methods and analysis of multiscale problems. (2017)

• Record Type: Book
• Further Information: Note: Alexandre L. Madureira.
• Authors: Madureira, Alexandre L
• Contents: Preface; Prerequisites and Contents; Acknowledgments; Contents; 1 Introductory Material and Finite Element Methods; 1.1 Introduction; 1.2 Galerkin Method; 1.3 Alternative Methods; 1.3.1 Residual Free Bubbles; 1.3.2 Multiscale Finite Element Method; 1.3.3 Variational Multiscale Method and Localization; 1.3.4 Heterogeneous Multiscale Method; 1.3.5 Hybrid Methods; 1.3.6 Extending the Discontinuous Enriched Method; 1.3.7 Stabilized Methods; 1.4 Conclusions; 2 One-Dimensional Singular Perturbed Problems; 2.1 Introduction; 2.2 Advection-Diffusion with Constant Coefficients. 2.2.1 The Problem and Its Finite Element Discretization2.2.2 So, What Goes Wrong?; 2.2.3 Matching Asymptotic Expansions; 2.3 A More General Singular Perturbed Second Order ODE; 2.3.1 Asymptotic Expansion; 2.3.2 Truncation Error Analysis; 2.4 Conclusions; 3 An Application in Neuroscience: Heterogeneous Cable Equation; 3.1 Introduction; 3.2 Classical Finite Element Method; 3.3 The Multiscale Finite Element Method; 3.4 Robustness of the Multiscale Method; 3.4.1 Low Diffusion Regime; 3.4.2 Large Number of Synapses; 3.5 Conclusions; 4 Two-Dimensional Reaction-Diffusion Equations; 4.1 Introduction. 4.2 The Continuous Problem4.2.1 Asymptotic Expansion; 4.2.2 Error Estimates for the Asymptotic Expansion; 4.2.3 Estimates for Non-smooth Domain; 4.3 Finite Element Approximations; 4.3.1 General Comments; 4.3.2 Enriching Finite Element Spaces; 4.3.2.1 New Enriched Choice; 4.3.2.2 Solving Local Problems; 4.3.2.3 A Numerical Test: Source Problem; 4.4 Conclusions; 5 Modeling PDEs in Domains with Rough Boundaries; 5.1 Introduction; 5.2 Asymptotic Expansion; 5.2.1 Asymptotic Expansion Definition; 5.2.2 The Boundary Corrector Problem; 5.3 Derivation of Wall-Laws. 5.4 A Multiscale Finite Element Method5.4.1 Method Definition; 5.4.2 Numerical Analysis; 5.4.2.1 Asymptotic Expansion of the Exact Solution; 5.5 Conclusions; 6 Partial Differential Equations with Oscillatory Coefficients; 6.1 Introduction; 6.2 Two-Scale Asymptotic Expansion; 6.2.1 Justifying the Asymptotic Expansion; 6.2.2 An Example; 6.3 A Finite Element Method Discretization; 6.3.1 What Goes Wrong?; 6.4 Multiscale Finite Element Method; 6.4.1 Definition of the Method; 6.4.2 Error Analysis; 6.4.3 An Extra Issue: High Contrast; 6.4.4 Further Comments; 6.5 Conclusions; References; Index.
• Publisher Details: Cham, Switzerland : Springer
• Publication Date: 2017
• Extent: 1 online resource, illustrations
• Subjects: 518.25; Mathematics; Finite element method; MATHEMATICS -- Essays; MATHEMATICS -- Pre-Calculus; MATHEMATICS -- Reference; MATHEMATICS -- Numerical Analysis; Finite element method; Mathematics -- Differential Equations; Mathematics -- Applied; Differential calculus & equations; Applied mathematics; Numerical analysis; Differential equations, partial; Mathematics -- Number Systems; Numerical analysis; Electronic books
• Languages: English
• ISBNs: 9783319508665; 3319508660
• Related ISBNs: 9783319508641; 3319508644
• Notes: Note: Includes bibliographical references and index.; Note: Vendor-supplied metadata.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.