{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:10:25Z","timestamp":1776838225610,"version":"3.51.2"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2007,5,25]],"date-time":"2007-05-25T00:00:00Z","timestamp":1180051200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2007,8,7]]},"DOI":"10.1007\/s00211-007-0086-x","type":"journal-article","created":{"date-parts":[[2007,5,24]],"date-time":"2007-05-24T10:31:41Z","timestamp":1180002701000},"page":"257-293","source":"Crossref","is-referenced-by-count":74,"title":["A fictitious domain approach to the numerical solution of PDEs in stochastic domains"],"prefix":"10.1007","volume":"107","author":[{"given":"Claudio","family":"Canuto","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomas","family":"Kozubek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2007,5,25]]},"reference":[{"key":"86_CR1","unstructured":"Babu\u0161ka, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. (to appear)"},{"key":"86_CR2","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1137\/S0036142902418680","volume":"42","author":"I. Babu\u0161ka","year":"2004","unstructured":"Babu\u0161ka I., Tempone R., Zouraris G.E. (2004). Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42: 800\u2013825","journal-title":"SIAM J. Numer. Anal."},{"key":"86_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s002110050301","volume":"78","author":"S. Bertoluzza","year":"1997","unstructured":"Bertoluzza S. (1997). Interior estimates for the wavelet Galerkin method. Numer. Math. 78: 1\u201320","journal-title":"Numer. Math."},{"key":"86_CR4","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F. Brezzi","year":"1991","unstructured":"Brezzi F., Fortin M. (1991). Mixed and Hybrid Finite Element Methods. Springer, New York"},{"key":"86_CR5","doi-asserted-by":"crossref","first-page":"385","DOI":"10.2307\/1969178","volume":"48","author":"R. Cameron","year":"1947","unstructured":"Cameron R., Martin W. (1947). The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48: 385\u2013392","journal-title":"Ann. Math."},{"key":"86_CR6","doi-asserted-by":"crossref","unstructured":"Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer., 1\u201349 (1998)","DOI":"10.1017\/S0962492900002804"},{"key":"86_CR7","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral Methods. Fundamentals in Single Domains","author":"C. Canuto","year":"2006","unstructured":"Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. (2006). Spectral Methods. Fundamentals in Single Domains. Springer, Berlin"},{"key":"86_CR8","doi-asserted-by":"crossref","first-page":"6359","DOI":"10.1016\/S0045-7825(01)00237-7","volume":"190","author":"M.K. Deb","year":"2001","unstructured":"Deb M.K., Babu\u0161ka I., Oden J.T. (2001). Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190: 6359\u20136372","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"86_CR9","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4778-4","volume-title":"Moduli of Smoothness","author":"Z. Ditzian","year":"1987","unstructured":"Ditzian Z., Totik V. (1987). Moduli of Smoothness. Springer, New York"},{"key":"86_CR10","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic Finite Elements\u2014A Spectral Approach","author":"R. Ghanem","year":"1991","unstructured":"Ghanem R., Spanos P.D. (1991). Stochastic Finite Elements\u2014A Spectral Approach. Springer, Berlin"},{"key":"86_CR11","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1023\/A:1019129717644","volume":"18","author":"T. Gerstner","year":"1998","unstructured":"Gerstner T., Griebel M. (1998). Numerical integration using sparse grids. Numer. Algorithms 18: 209\u2013232","journal-title":"Numer. Algorithms"},{"key":"86_CR12","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/BF03167240","volume":"12","author":"V. Girault","year":"1995","unstructured":"Girault V., Glowinski R. (1995). Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Ind. Appl. Math. 12: 487\u2013514","journal-title":"Japan J. Ind. Appl. Math."},{"key":"86_CR13","first-page":"159","volume-title":"Boundary Value Problems for Partial Differential Equations and Applications","author":"R. Glowinski","year":"1993","unstructured":"Glowinski R., Pan T.S., P\u00e9riaux J. (1993). A least squares\/fictitious domain method for mixed problems and Neumann problems. In: Lions, J.L., Baiocchi, C. (eds) Boundary Value Problems for Partial Differential Equations and Applications, pp 159\u2013178. Masson, Paris"},{"key":"86_CR14","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/0045-7825(94)90135-X","volume":"111","author":"R. Glowinski","year":"1994","unstructured":"Glowinski R., Pan T., Periaux J. (1994). A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111: 283\u2013303","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"86_CR15","first-page":"1","volume":"8","author":"J. Haslinger","year":"2000","unstructured":"Haslinger J., Kozubek T. (2000). A fictitious domain approach for a class of Neumann boundary value problems with applications in shape optimization. East-West J Numer. Math. 8: 1\u201326","journal-title":"East-West J Numer. Math."},{"key":"86_CR16","doi-asserted-by":"crossref","unstructured":"Haslinger, J., Kozubek, T., Kucera, R., Peichl, G.: Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach. Numer. Linear Algebra Appl. (2007) (to appear)","DOI":"10.1002\/nla.550"},{"key":"86_CR17","first-page":"231","volume":"26","author":"J. Haslinger","year":"2003","unstructured":"Haslinger J., Kozubek T., Kunisch K., Peichl G. (2003). Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. COAP 26: 231\u2013251","journal-title":"COAP"},{"key":"86_CR18","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718690","volume-title":"Introduction to Shape Optimization, Theory, Approximation, and Computation","author":"J. Haslinger","year":"2003","unstructured":"Haslinger J., M\u00e4kinen R.A.E. (2003). Introduction to Shape Optimization, Theory, Approximation and Computation. SIAM, Philadelphia"},{"key":"86_CR19","doi-asserted-by":"crossref","unstructured":"Hosden, S., Walters, R.W., Perez, R.: A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. AIAA Paper 2006-0891","DOI":"10.2514\/6.2006-891"},{"key":"86_CR20","volume-title":"The Stochastic Finite Element Method. Basic Perturbation Technique and Computer Implementation","author":"M. Kleiber","year":"1992","unstructured":"Kleiber M., Hien T.D. (1992). The Stochastic Finite Element Method. Basic Perturbation Technique and Computer Implementation. Wiley, Chichester"},{"key":"86_CR21","unstructured":"Kucera, R., Kozubek, T., Haslinger, J.: On solving non-symmetric saddle-point systems arising from fictitious domain approaches. In: Proceedings of PANM06, Prague, Czech Republic, pp. 165\u2013171 (2006)"},{"key":"86_CR22","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9464-8","volume-title":"Probability Theory","author":"M. Lo\u00e8ve","year":"1977","unstructured":"Lo\u00e8ve M. (1977). Probability Theory. Springer-Verlag, Berlin"},{"key":"86_CR23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF02874673","volume":"55","author":"G. Mastroianni","year":"2006","unstructured":"Mastroianni G., Monegato G. (2006). Truncated approximation processes on unbounded intervals and their applications. Rend. Circ. Mat. Palermo 55: 123\u2013139","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"86_CR24","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s11075-004-2866-z","volume":"38","author":"I. Mathelin","year":"2005","unstructured":"Mathelin I., Hussaini M.Y., Zang T.A. (2005). Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38: 209\u2013236","journal-title":"Numer. Algorithms"},{"key":"86_CR25","doi-asserted-by":"crossref","unstructured":"Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, pp. 1295\u20131331 in [28]","DOI":"10.1016\/j.cma.2004.05.027"},{"key":"86_CR26","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1093\/imanum\/dri045","volume":"26","author":"M.S. Mommer","year":"2006","unstructured":"Mommer M.S. (2006). A smoothness preserving fictitious domain method for elliptic boundary-value problems. IMA J. Numer. Anal. 26: 503\u2013524","journal-title":"IMA J. Numer. Anal."},{"key":"86_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03620-4","volume-title":"Stochastic Differential Equations\u2014An Introduction with Applications","author":"B. Oksendal","year":"1998","unstructured":"Oksendal B. (1998). Stochastic Differential Equations\u2014An Introduction with Applications. Springer, Berlin"},{"key":"86_CR28","doi-asserted-by":"crossref","unstructured":"Schu\u00ebller, G.I. (ed.): Computational Methods in Stochastic Mechanics and Reliability Analysis. Special Issue 12\u201316, Comput. Methods Appl. Mech. Eng. 194 (2005)","DOI":"10.1016\/j.cma.2004.08.001"},{"key":"86_CR29","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s00607-003-0024-4","volume":"71","author":"Ch. Schwab","year":"2003","unstructured":"Schwab Ch., Todor R.A. (2003). Sparse finite elements for stochastic elliptic problems-higher order moments. Computing 71: 43\u201363","journal-title":"Computing"},{"key":"86_CR30","unstructured":"Todor, R.A., Schwab, Ch.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. Research Rep. No. 2006-05, Seminar f\u00fcr Angewandte Mathematik, ETH Zurich"},{"key":"86_CR31","doi-asserted-by":"crossref","unstructured":"Walters, R.W.: Towards stochastic fluid mechanics via polynomial chaos. AIAA Paper 2003-0413","DOI":"10.2514\/6.2003-413"},{"key":"86_CR32","doi-asserted-by":"crossref","first-page":"897","DOI":"10.2307\/2371268","volume":"60","author":"N. Wiener","year":"1938","unstructured":"Wiener N. (1938). The homogeneous chaos. Am. J. Math. 60: 897\u2013936","journal-title":"Am. J. Math."},{"key":"86_CR33","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1137\/S1064827501387826","volume":"24","author":"D. Xiu","year":"2002","unstructured":"Xiu D., Karniadakis G.E. (2002). The Wiener\u2013Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24: 619\u2013644","journal-title":"SIAM J. Sci. Comput."},{"key":"86_CR34","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0021-9991(03)00092-5","volume":"187","author":"D. Xiu","year":"2003","unstructured":"Xiu D., Karniadakis G.E. (2003). Modeling uncertainty in flow simulation via generalized polynomial chaos. J. Comput. Phys. 187: 137\u2013167","journal-title":"J. Comput. Phys."},{"key":"86_CR35","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1137\/040613160","volume":"28","author":"D. Xiu","year":"2006","unstructured":"Xiu D., Tartakovsky D.M. (2006). Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28: 1167\u20131185","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-007-0086-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-007-0086-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-007-0086-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,14]],"date-time":"2021-08-14T03:29:36Z","timestamp":1628911776000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-007-0086-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5,25]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,8,7]]}},"alternative-id":["86"],"URL":"https:\/\/doi.org\/10.1007\/s00211-007-0086-x","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,5,25]]}}}