Computer-assisted proofs for semilinear elliptic boundary value problems

For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a “close” and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach’s fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

[1]  K. Rektorys.  Variational Methods in Mathematics, Science and Engineering , 1977 .

[2]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Konstantin Mischaikow,et al.  Rigorous Numerics for Global Dynamics: A Study of the Swift-Hohenberg Equation , 2005, SIAM J. Appl. Dyn. Syst..

[5]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[6]  M. Nakao.  Solving Nonlinear Elliptic Problems with Result Verification Using an H -1 Type Residual Iteration , 1993 .

[7]  M. Plum.  Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems , 1995 .

[8]  S. Zimmermann,et al.  Variational Bounds to Eigenvalues of Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum , 1995 .

[9]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[10]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[11]  Christian Wieners,et al.  Enclosure for the Biharmonic Equation , 2005, Algebraic and Numerical Algorithms and Computer-assisted Proofs.

[12]  M. Plum,et al.  A computer‐assisted instability proof for the Orr‐Sommerfeld equation with Blasius profile , 2004 .

[13]  W. Walter.  Differential and Integral Inequalities , 1970 .

[14]  H. Behnke,et al.  Inclusion of eigenvalues of general eigenvalue problems for matrices , 1988 .

[15]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[16]  H. Bauer.  Wahrscheinlichkeitstheorie und Grundzuge der Maßtheorie , 1968 .

[17]  J. McWhirter.  Variational Methods in Mathematics, Science and Engineering , 1978 .

[18]  Michael Plum,et al.  Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems , 1992 .

[19]  G. Corliss,et al.  C-Xsc: A C++ Class Library for Extended Scientific Computing , 1993 .

[20]  P. J. McKenna,et al.  A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam , 2006 .

[21]  N. Lehmann.  Optimale Eigenwerteinschließungen , 1963 .

[22]  Y. Choi,et al.  A mountain pass method for the numerical solution of semilinear elliptic problems , 1993 .

[23]  M. Nakao,et al.  An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness , 1999 .

[24]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[25]  Michael Plum,et al.  Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof , 2000 .

[26]  M. Nakao,et al.  Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element , 1995 .

[27]  Michael Plum,et al.  A computer‐assisted instability proof for the Orr‐Sommerfeld problem with Poiseuille flow , 2009 .

[28]  Lothar Collatz,et al.  Aufgaben monotoner Art , 1952 .

[29]  M. Plum,et al.  New solutions of the Gelfand problem , 2002 .