Klaus Böhmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Chapter 2 summarizes general linear, special semilinear, semilinear, quasilinear, and fully nonlinear elliptic differential equations and systems of order 2m, m ≥ 1, e.g. the above equations. ...
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Chapter 2 summarizes general linear, special semilinear, semilinear, quasilinear, and fully nonlinear elliptic differential equations and systems of order 2m, m ≥ 1, e.g. the above equations. Essential are existence, uniqueness, and regularity of their solutions and linearization. Many important arguments for linearization are discussed. It is assumed that the derivative of the nonlinear operator, evaluated in the exact (isolated) solution, is boundedly invertible, closely related to the numerically necessary condition of a (locally) well-conditioned problem. Bifurcation problems are delayed to the next book; ill-conditioned problems are not considered. Linearization is applicable to nearly all nonlinear elliptic problems. Its bounded invertibility yields the Fredholm alternative and the stability of space discretization methods. Some nonlinear, monotone problems exclude linearization.Less
Chapter 2 summarizes general linear, special semilinear, semilinear, quasilinear, and fully nonlinear elliptic differential equations and systems of order 2m, m ≥ 1, e.g. the above equations. Essential are existence, uniqueness, and regularity of their solutions and linearization. Many important arguments for linearization are discussed. It is assumed that the derivative of the nonlinear operator, evaluated in the exact (isolated) solution, is boundedly invertible, closely related to the numerically necessary condition of a (locally) well-conditioned problem. Bifurcation problems are delayed to the next book; ill-conditioned problems are not considered. Linearization is applicable to nearly all nonlinear elliptic problems. Its bounded invertibility yields the Fredholm alternative and the stability of space discretization methods. Some nonlinear, monotone problems exclude linearization.
Klaus Böhmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of ...
More
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C1 FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ2 for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.Less
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C1 FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ2 for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.
Klaus Boehmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important ...
More
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important examples. This is the first and only book, proving in a systematic and unifying way, stability and convergence results and methods for solving nonlinear discrete equations via discrete Newton methods for the different numerical methods for all these problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. This is examplified for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference and wavelet methods. The proof of stability for nonconforming methods employs the anticrime operator as an essential tool. For all these methods approximate evaluation of the discrete equations, and eigenvalue problems are discussed. The numerical methods are based upon analytic results for this wide class of problems, guaranteeing existence, uniqueness and regularity of the exact solutions. In the next book, spectral, mesh‐free methods and convergence for bifurcation and center manifolds for all these combinations are proved. Specific long open problems, solved here, are numerical methods for fully nonlinear elliptic problems, wavelet and mesh‐free methods for nonlinear problems, and more general nonlinear boundary conditions. Adaptivity is discussed for finite element and wavelet methods with totally different techniques.Less
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important examples. This is the first and only book, proving in a systematic and unifying way, stability and convergence results and methods for solving nonlinear discrete equations via discrete Newton methods for the different numerical methods for all these problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. This is examplified for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference and wavelet methods. The proof of stability for nonconforming methods employs the anticrime operator as an essential tool. For all these methods approximate evaluation of the discrete equations, and eigenvalue problems are discussed. The numerical methods are based upon analytic results for this wide class of problems, guaranteeing existence, uniqueness and regularity of the exact solutions. In the next book, spectral, mesh‐free methods and convergence for bifurcation and center manifolds for all these combinations are proved. Specific long open problems, solved here, are numerical methods for fully nonlinear elliptic problems, wavelet and mesh‐free methods for nonlinear problems, and more general nonlinear boundary conditions. Adaptivity is discussed for finite element and wavelet methods with totally different techniques.
