{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:11:09Z","timestamp":1680261069737},"reference-count":40,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"Abstract<\/jats:title>\n In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh \n \n \n \n \n \ud835\udcaf<\/m:mi>\n H<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n D<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{T}_{H}(D)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with small scale of sparse collocation points \n \n \n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n L<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\eta(L,N)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and solve a linearized equations on a fine mesh \n \n \n \n \n \ud835\udcaf<\/m:mi>\n h<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n D<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{T}_{h}(D)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> using large scale of sparse collocation points \n \n \n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u2113<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\eta(\\ell,N)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (where \n \n \n \n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n L<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u2113<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\eta(L,N),\\eta(\\ell,N)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are the numbers of sparse grid with respect to different levels \n \n \n \n L<\/m:mi>\n ,<\/m:mo>\n \u2113<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {L,\\ell}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in N<\/jats:italic> dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when \n \n \n \n \n h<\/m:mi>\n \u2248<\/m:mo>\n \n H<\/m:mi>\n 3<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n ,<\/m:mo>\n \n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u2113<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2248<\/m:mo>\n \n \n (<\/m:mo>\n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n L<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n 3<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {{h\\approx H^{3},\\eta(\\ell,N)\\approx(\\eta(L,N))^{3}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the novel two-level discretization method achieves the same convergence accuracy in norm \n \n \n \n \u2225<\/m:mo>\n \u22c5<\/m:mo>\n \n \u2225<\/m:mo>\n \n \n \n \n \u2112<\/m:mi>\n \u03c1<\/m:mi>\n 2<\/m:mn>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u0393<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2297<\/m:mo>\n \n \u2112<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n D<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n <\/m:math>\n \n {\\|\\cdot\\|_{\\mathcal{L}_{\\rho}^{2}(\\Gamma)\\otimes\\mathcal{L}^{2}(D)}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (\n \n \n \n \n \u2112<\/m:mi>\n \u03c1<\/m:mi>\n 2<\/m:mn>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u0393<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{L}_{\\rho}^{2}(\\Gamma)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is the weighted \n \n \n \n \u2112<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {\\mathcal{L}^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> space with \u03c1 a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with \n \n \n \n \n \ud835\udcaf<\/m:mi>\n h<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n D<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{T}_{h}(D)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and large scale collocation points \n \n \n \n \u03b7<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u2113<\/m:mi>\n ,<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\eta(\\ell,N)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in random spaces.<\/jats:p>","DOI":"10.1515\/cmam-2017-0025","type":"journal-article","created":{"date-parts":[[2017,7,28]],"date-time":"2017-07-28T10:01:43Z","timestamp":1501236103000},"page":"165-179","source":"Crossref","is-referenced-by-count":0,"title":["A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation"],"prefix":"10.1515","volume":"18","author":[{"given":"Luoping","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics , Southwest Jiaotong University , Chengdu 611756 , P. R. China"}]},{"given":"Yanping","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematical Science , South China Normal University , Guangzhou 510631 , P. R. China"}]},{"given":"Xiong","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Lingnan Normal University , Zhanjiang 524048 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2017,7,28]]},"reference":[{"key":"2023033109580926599_j_cmam-2017-0025_ref_001_w2aab3b7e2794b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"G. Adomian,\nNonlinear Stochastic Systems: Theory and Application to Physics,\nSpringer, Netherlands, 1989.","DOI":"10.1007\/978-94-009-2569-4"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_002_w2aab3b7e2794b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"A. Ammi and M. Marion,\nNonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier\u2013Stokes equations,\nNumer. 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Eng. 57 (2003), no. 2, 193\u2013209.","DOI":"10.1002\/nme.668"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_009_w2aab3b7e2794b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Y. Chen, H. Liu and S. Liu,\nAnalysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods,\nInt. J. Numer. Meth. Eng. 69 (2007), no. 2, 408\u2013422.","DOI":"10.1002\/nme.1775"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_010_w2aab3b7e2794b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"C. Chien and B. Jeng,\nA two-grid discretization scheme for semilinear elliptic eigenvalue problems,\nSIAM J. Sci. Comput. 27 (2006), no. 4, 1287\u20131304.","DOI":"10.1137\/030602447"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_011_w2aab3b7e2794b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"C. Dawson and M. Wheeler,\nTwo-grid methods for mixed finite element approximations of nonlinear parabolic equations,\nContemp. Math. 180 (1994), 191\u2013191.","DOI":"10.1090\/conm\/180\/01971"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_012_w2aab3b7e2794b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"C. Dawson, M. Wheeler and C. Woodward,\nA two-grid finite difference scheme for nonlinear parabolic equations,\nSIAM J. Numer. Anal. 35 (1998), no. 2, 435\u2013452.","DOI":"10.1137\/S0036142995293493"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_013_w2aab3b7e2794b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"T. Gerstner and M. Griebel,\nNumerical integration using sparse grids,\nNumer. Algorithms 18 (1998), no. 3, 209.","DOI":"10.1023\/A:1019129717644"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_014_w2aab3b7e2794b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"R. Ghanem and P. Spanos,\nStochastic Finite Elements: A Spectral Approach,\nSpringer, New York, 1991.","DOI":"10.1007\/978-1-4612-3094-6"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_015_w2aab3b7e2794b1b6b1ab2ac15Aa","unstructured":"V. Girault and J. Lions,\nTwo-grid finite-element schemes for the steady Navier\u2013Stokes problem in polyhedra,\nPort. Math. 58 (2001), no. 1, 25\u201358."},{"key":"2023033109580926599_j_cmam-2017-0025_ref_016_w2aab3b7e2794b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"S. Hosder, R. Walters and R. Perez,\nA non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations,\nProceedings of the 44th AIAA Aerospace Sciences Meeting and Exibit (Reno 2006),\nAIAA, Reston (2006), 1\u201319.","DOI":"10.2514\/6.2006-891"},{"key":"2023033109580926599_j_cmam-2017-0025_ref_017_w2aab3b7e2794b1b6b1ab2ac17Aa","unstructured":"A. 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