*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/9780199577040.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A new general discretization theory unifies the generalized Petrov-Galerkin method and one of the classical methods. Linearization is a main tool: the derivative of the operator in the exact solution ...
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A new general discretization theory unifies the generalized Petrov-Galerkin method and one of the classical methods. Linearization is a main tool: the derivative of the operator in the exact solution has to be boundedly invertible. For quasilinear problems, in Sobolev spaces Wm'p(Ω), with 2 ≤ p 〈 ∞, this chapter obtains stability and convergence results with respect to discrete Hm(Ω) norms. This is complemented by the monotone approach for 1 ≤ p 〈 ∞ with Wm'p(Ω) convergence. Our approach allows a unified proof for stability, convergence and Fredholm results for the discrete solutions and their computation. A few well-known basic concepts from functional analysis and approximation theory are combined: coercive bilinear forms or monotone operators, their compact perturbations, interpolation, best approximation and inverse estimates for approximating spaces yield the classical “consistency and stability imply convergence”. The mesh independence principle is the key for an efficient solution for all discretizations of all nonlinear problems considered here.Less

A new general discretization theory unifies the generalized Petrov-Galerkin method and one of the classical methods. Linearization is a main tool: the derivative of the operator in the exact solution has to be boundedly invertible. For quasilinear problems, in Sobolev spaces *W*^{m'p}(Ω), with 2 ≤ *p* 〈 ∞, this chapter obtains stability and convergence results with respect to discrete *H*^{m}(Ω) norms. This is complemented by the monotone approach for 1 ≤ *p* 〈 ∞ with *W*^{m'p}(Ω) convergence. Our approach allows a unified proof for stability, convergence and Fredholm results for the discrete solutions and their computation. A few well-known basic concepts from functional analysis and approximation theory are combined: coercive bilinear forms or monotone operators, their compact perturbations, interpolation, best approximation and inverse estimates for approximating spaces yield the classical “consistency and stability imply convergence”. The *mesh independence principle* is the key for an efficient solution for all discretizations of all nonlinear problems considered here.

*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 ...
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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.

*D. E. Edmunds and W. D. Evans*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0009
- Subject:
- Mathematics, Pure Mathematics

In this chapter, various essential spectra are studied. For a closed operator in a Banach space, a number of different sets have been used for the essential spectrum, the sets being identical for a ...
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In this chapter, various essential spectra are studied. For a closed operator in a Banach space, a number of different sets have been used for the essential spectrum, the sets being identical for a self-adjoint operator in a Hilbert space. As well as the essential spectra, the changes that occur when the operator is perturbed are discussed. Constant-coefficient differential operators are studied in detail.Less

In this chapter, various essential spectra are studied. For a closed operator in a Banach space, a number of different sets have been used for the essential spectrum, the sets being identical for a self-adjoint operator in a Hilbert space. As well as the essential spectra, the changes that occur when the operator is perturbed are discussed. Constant-coefficient differential operators are studied in detail.

*D. E. Edmunds and W. D. Evans*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0011
- Subject:
- Mathematics, Pure Mathematics

This chapter is devoted to the study of the Schrödinger operator −Δ + q with q real, and, in particular, the distribution of its eigenvalues. A general result is established on an open subset Ω of ...
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This chapter is devoted to the study of the Schrödinger operator −Δ + q with q real, and, in particular, the distribution of its eigenvalues. A general result is established on an open subset Ω of Rn using the Max–Min Principle and covering families of congruent cubes for the Dirichlet problem and a Whitney covering for the Neumann problem. The Cwikel–Lieb–Rosenbljum inequality is proved for q in Ln/2(Rn).Less

This chapter is devoted to the study of the Schrödinger operator −Δ + *q* with *q* real, and, in particular, the distribution of its eigenvalues. A general result is established on an open subset Ω of R^{n} using the Max–Min Principle and covering families of congruent cubes for the Dirichlet problem and a Whitney covering for the Neumann problem. The Cwikel–Lieb–Rosenbljum inequality is proved for *q* in L^{n/2}(R^{n}).