Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation

DOI10.1007/s13160-014-0156-2zbMath1335.65090OpenAlexW2093708554MaRDI QIDQ2257618

Kazuaki Tanaka, Shin'ichi Oishi, Xuefeng Liu, Akitoshi Takayasu

Publication date: 25 February 2015

Published in: Japan Journal of Industrial and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s13160-014-0156-2

zbMATH Keywords

finite element methodinverse operatornumerical exampleseigenvalue problemnumerical verificationelliptic operatorinverse norm estimation

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Estimates of eigenvalues in context of PDEs (35P15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) General theory of partial differential operators (47F05) General (adjoints, conjugates, products, inverses, domains, ranges, etc.) (47A05) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25)

Related Items (11)

Computational Lower Bounds of the Maxwell Eigenvalues ⋮ Mixed methods and lower eigenvalue bounds ⋮ Rigorous Numerical Enclosures for Positive Solutions of Lane–Emden’s Equation with Sub-Square Exponents ⋮ A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator ⋮ A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations ⋮ Sharp numerical inclusion of the best constant for embedding \(H_0^1(\Omega) \hookrightarrow L^p(\Omega)\) on bounded convex domain ⋮ Numerical verification method for positive solutions of elliptic problems ⋮ Numerical verification for asymmetric solutions of the Hénon equation on bounded domains ⋮ Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems ⋮ A posteriori verification of the positivity of solutions to elliptic boundary value problems ⋮ Numerical verification of positiveness for solutions to semilinear elliptic problems

Uses Software

• INTLAB

Cites Work

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• Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse
• Computer-assisted proofs for semilinear elliptic boundary value problems
• Estimation of interpolation error constants for the \(P_0\) and \(P_1\) triangular finite elements
• The calculation of guaranteed bounds for eigenvalues using complementary variational principles
• Numerical verification of existence and inclusion of solutions for nonlinear operator equations
• Guaranteed high-precision estimation for \(P_0\) interpolation constants on triangular finite elements
• A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems
• A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam
• Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape

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