{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:52:35Z","timestamp":1776837155934,"version":"3.51.2"},"reference-count":50,"publisher":"American Mathematical Society (AMS)","issue":"344","license":[{"start":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T00:00:00Z","timestamp":1720137600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincar\u00e9 operator is used to project the eigenvalue problem on the domain\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n onto the nonlinear eigenvalue subproblem on\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Four different strategies are proposed to build the hierarchical subspace\n \n \n \n \n U<\/mml:mi>\n \n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n U_{k+1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace\n \n \n \n \n U<\/mml:mi>\n \n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n U_{k+1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n c<\/mml:mi>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n T<\/mml:mi>\n \n h<\/mml:mi>\n ,<\/mml:mo>\n H<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\gamma =1-c_{0}T_{h,H}^{-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n c<\/mml:mi>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n c_{0}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a constant independent of the fine mesh size\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the coarse mesh size\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and jumps of the coefficients; whereas\n \n \n \n \n T<\/mml:mi>\n \n h<\/mml:mi>\n ,<\/mml:mo>\n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n T_{h,H}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.\n <\/p>","DOI":"10.1090\/mcom\/3860","type":"journal-article","created":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T09:34:21Z","timestamp":1681896861000},"page":"2655-2684","source":"Crossref","is-referenced-by-count":3,"title":["A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems"],"prefix":"10.1090","volume":"92","author":[{"given":"Wenbin","family":"Chen","sequence":"first","affiliation":[]},{"given":"Nian","family":"Shao","sequence":"additional","affiliation":[]},{"given":"Xuejun","family":"Xu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,7,5]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"1249","DOI":"10.1137\/0724082","article-title":"Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues","volume":"24","author":"Babu\u0161ka, I.","year":"1987","journal-title":"SIAM J. 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