{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:30:19Z","timestamp":1787337019302,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>We consider a preconditioning strategy for finite differences (FD) matrix sequences $\\{A_n(a,\\Omega)\\}_n$ discretizing the elliptic problem $$ \\label{eq:pde-abs} \\left\\{ \\begin{array}{l} A_{a} u \\equiv (-)^k\\nabla^k[a(x)\\nabla^k u(x)] =f(x),\\quad \\quad x\\in \\Omega, \\\\ \\ \\\\ \\displaystyle \\left({\\partial^s \\over \\partial \\nu^s} u(x)\\right)_{|\\partial \\Omega} \\equiv 0, \\quad \\quad s=0,\\ldots,k-1, \\end{array} \\right. $$ with $\\Omega$ being a plurirectangle of ${\\bf R}^d$, with $a(x)$ being a uniformly positive (nonnegative) Riemann integrable function, and $\\nu$ denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG)--like methods, we consider the preconditioning sequence $\\{P_n(a,\\Omega)\\}_n$, $P_n(a,\\Omega):=\\tilde D_n^{1\/2}(a,\\Omega)A_n(1,\\Omega) \\tilde D_n^{1\/2}(a,\\Omega)$, where $\\tilde D_n(a,\\Omega)$ is the suitable scaled main diagonal of $A_n(a,\\Omega)$. Using embedding arguments and projection matrices, under the mild assumptions on $a(x)$, we show the weak clustering at the unity of the corresponding preconditioned sequence. If $a(x)$ is regular enough, then the preconditioned sequence shows a strong clustering at the unity so that the sequence $\\{P_n(a,\\Omega)\\}_n$ turns out to be a superlinear preconditioning sequence for $\\{A_n(a,\\Omega)\\}_n$. The computational interest is due to the fact that the solution of a linear system with coefficient matrix $A_n(a,\\Omega)$ is reduced to computations involving diagonals and multilevel structures $\\{A_n(1,\\Omega)\\}_n$ with banded pattern. In turn, the matrix $A_n(1,\\Omega)$ can be reinterpreted as a projection of a multilevel banded Toeplitz matrix for which we use multigrid strategies. Some numerical experimentations confirm the efficiency of the discussed proposal and its strong superiority with respect to existing techniques in the case of semielliptic problems.<\/jats:p>","DOI":"10.1137\/s0895479802416058","type":"journal-article","created":{"date-parts":[[2003,11,7]],"date-time":"2003-11-07T11:17:43Z","timestamp":1068203863000},"page":"152-164","source":"Crossref","is-referenced-by-count":5,"title":["Superlinear Preconditioners for Finite Differences Linear Systems"],"prefix":"10.1137","volume":"25","author":[{"given":"Stefano Serra","family":"Capizzano","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cristina Tablino","family":"Possio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"O. Axelsson and V. Barker,\n                      Finite Element Solution of Boundary Value Problems, Theory and Computation\n                      , Academic Press, New York, 1984.","DOI":"10.1016\/B978-0-12-068780-0.50011-X"},{"key":"R2","unstructured":"B. Beckermann and S. Serra Capizzano,\n                      Spectral Distributions of Finite Element Matrix Sequences\n                      , manuscript, 2002."},{"key":"R3","unstructured":"D. Bertaccini, G. Golub, S. Serra Capizzano, and C. Tablino Possio,\n                      Preconditioned HSS Method for the Solution of Non\u2010Hermitian Positive Definite Linear Systems\n                      , Tech. Report SCCM\u201002\u201010, Stanford University, Stanford, CA, 2002."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)00015-9"},{"key":"R5","unstructured":"J. Bramble,\n                      Multigrid Methods\n                      , Pitman Res. Notes in Math. Ser. 294, Longman Scientific, Harlow, UK, 1993."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0707049"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0708066"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595293831"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"P. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , Classics in Appl. Math. 40, SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0710092"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/1012045"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594271512"},{"key":"R13","unstructured":"A. Frangioni and S. Serra Capizzano,\n                      Spectral Analysis and Preconditioning of Local Graph Matrices and Application to FD Linear Systems\n                      , manuscript, 2002."},{"key":"R14","unstructured":"C. Johnson,\n                      Numerical Solutions of Partial Differential Equations by the Finite Elements Methods\n                      , Cambridge University Press, Cambridge, UK, 1988."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Multigrid Methods and Applications\n                      , Springer\u2010Verlag, Berlin, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050142"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)80014-1"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050400"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898338366"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100331"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00504-9"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(99)00022-1"},{"key":"R23","first-page":"55","volume":"11","author":"Serra Capizzano Stefano","year":"2000","journal-title":"Electron. Trans. Numer. Anal."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"S. Serra Capizzano, C. Tablino Possio, Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs, Contemp. Math., Vol. 281, Amer. Math. Soc., Providence, RI, 2001, 295\u20133172002i:65116","DOI":"10.1090\/conm\/281\/04664"},{"key":"R25","unstructured":"S. Serra Capizzano and C. Tablino Possio,\n                      Structured preconditioning of optimal preconditioners for 2D collocation linear systems\n                      , in Structured Matrices: Recent Developments in Theory and Computations, D. Bini, E. Tyrtyshnikov, and P. Yalamov, eds., Nova Science Publishers, Hauppauge, NY, 2001, pp. 191\u2013204."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014088011253"},{"key":"R27","first-page":"1","volume":"16","author":"Serra Capizzano S.","year":"2003","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R28","unstructured":"S. Serra Capizzano and C. Tablino Possio,\n                      Analysis of preconditioning strategies for collocation linear systems\n                      , Linear Algebra Appl., to appear."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/1019071"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511812545"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479802416058","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:41:00Z","timestamp":1787334060000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479802416058"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479802416058"],"URL":"https:\/\/doi.org\/10.1137\/s0895479802416058","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}