LEADER 05452nam 2200517 450 001 996490271503316 005 20231110225229.0 010 $a3-031-12616-5 035 $a(CKB)5840000000091739 035 $a(MiAaPQ)EBC7101860 035 $a(Au-PeEL)EBL7101860 035 $a(PPN)264952634 035 $a(EXLCZ)995840000000091739 100 $a20230223d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aNumerical methods for mixed finite element problems $eapplications to incompressible materials and contact problems /$fJean Deteix, Thierno Diop and Michel Fortin 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (119 pages) 225 1 $aLecture Notes in Mathematics ;$vv.2318 311 $a3-031-12615-7 320 $aIncludes bibliographical references and index. 327 $aIntro -- Contents -- 1 Introduction -- 2 Mixed Problems -- 2.1 Some Reminders About Mixed Problems -- 2.1.1 The Saddle Point Formulation -- 2.1.2 Existence of a Solution -- 2.1.3 Dual Problem -- 2.1.4 A More General Case: A Regular Perturbation -- 2.1.5 The Case -- 2.2 The Discrete Problem -- 2.2.1 Error Estimates -- 2.2.2 The Matricial Form of the Discrete Problem -- 2.2.3 The Discrete Dual Problem: The Schur Complement -- 2.3 Augmented Lagrangian -- 2.3.1 Augmented or Regularised Lagrangians -- 2.3.2 Discrete Augmented Lagrangian in Matrix Form -- 2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem -- 2.3.4 Augmented Lagrangian: An Iterated Penalty -- 3 Iterative Solvers for Mixed Problems -- 3.1 Classical Iterative Methods -- 3.1.1 Some General Points -- Linear Algebra and Optimisation -- Norms -- Krylov Subspace -- Preconditioning -- 3.1.2 The Preconditioned Conjugate Gradient Method -- 3.1.3 Constrained Problems: Projected Gradient and Variants -- Equality Constraints: The Projected Gradient Method -- Inequality Constraints -- Positivity Constraints -- Convex Constraints -- 3.1.4 Hierarchical Basis and Multigrid Preconditioning -- 3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm -- The Generalised Conjugate Residual Method -- The Left Preconditioning -- The Right Preconditioning -- The Gram-Schmidt Algorithm -- GCR for Mixed Problems -- 3.2 Preconditioners for the Mixed Problem -- 3.2.1 Factorisation of the System -- Solving Using the Factorisation -- 3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm -- The Uzawa Algorithm -- 3.2.3 The General Preconditioned Algorithm -- 3.2.4 Augmented Lagrangian as a Perturbed Problem -- 4 Numerical Results: Cases Where Q= Q -- 4.1 Mixed Laplacian Problem -- 4.1.1 Formulation of the Problem. 327 $a4.1.2 Discrete Problem and Classic Numerical Methods -- The Augmented Lagrangian Formulation -- 4.1.3 A Numerical Example -- 4.2 Application to Incompressible Elasticity -- 4.2.1 Nearly Incompressible Linear Elasticity -- 4.2.2 Neo-Hookean and Mooney-Rivlin Materials -- Mixed Formulation for Mooney-Rivlin Materials -- 4.2.3 Numerical Results for the Linear Elasticity Problem -- 4.2.4 The Mixed-GMP-GCR Method -- Approximate Solver in u -- 4.2.5 The Test Case -- Number of Iterations and Mesh Size -- Comparison of the Preconditioners of Sect.3.2 -- Effect of the Solver in u -- 4.2.6 Large Deformation Problems -- Neo-Hookean Material -- Mooney-Rivlin Material -- 4.3 Navier-Stokes Equations -- 4.3.1 A Direct Iteration: Regularising the Problem -- 4.3.2 A Toy Problem -- 5 Contact Problems: A Case Where Q?Q -- 5.1 Imposing Dirichlet's Condition Through a Multiplier -- 5.1.1 and Its Dual -- 5.1.2 A Steklov-Poincaré operator -- Using This as a Solver -- 5.1.3 Discrete Problems -- The Matrix Form and the Discrete Schur Complement -- 5.1.4 A Discrete Steklov-Poincaré Operator -- 5.1.5 Computational Issues, Approximate Scalar Product -- Simplified Forms of the ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S script upper P Subscript h) /StPNE pdfmark [/StBMC pdfmarkSPhps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Operator and Preconditioning -- 5.1.6 The Formulation -- The Choice of h -- 5.1.7 A Toy Model for the Contact Problem -- The Discrete Formulation -- The Active Set Strategy -- 5.2 Sliding Contact -- 5.2.1 The Discrete Contact Problem -- Contact Status -- 5.2.2 The Algorithm for Sliding Contact -- A Newton Method -- The Active Set Strategy -- 5.2.3 A Numerical Example of Contact Problem -- 6 Solving Problems with More Than One Constraint -- 6.1 A Model Problem -- 6.2 Interlaced Method -- 6.3 Preconditioners Based on Factorisation. 327 $a6.3.1 The Sequential Method -- 6.4 An Alternating Procedure -- 7 Conclusion -- Bibliography -- Index. 410 0$aLecture Notes in Mathematics 606 $aFinite element method 606 $aMètode dels elements finits$2thub 608 $aLlibres electrònics$2thub 615 0$aFinite element method. 615 7$aMètode dels elements finits 676 $a620.00151535 700 $aDeteix$b Jean$01258294 702 $aDiop$b Thierno$c(Mathematician), 702 $aFortin$b Michel 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996490271503316 996 $aNumerical methods for mixed finite element problems$93035942 997 $aUNISA