{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T00:17:55Z","timestamp":1684541875026},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Our goal is to present an elementary approach to the\nanalysis and programming of sparse grid finite element methods. This\nfamily of schemes can compute accurate solutions to partial\ndifferential equations, but using far fewer degrees of freedom than\ntheir classical counterparts.\nAfter a brief discussion of the\nclassical Galerkin finite element method with bilinear elements, we\ngive a short analysis of what is probably the simplest sparse grid\nmethod: the two-scale technique of Lin et\nal. [14]. We\nthen demonstrate how to extend this to a <jats:italic>multiscale<\/jats:italic> sparse\ngrid method which, up to choice of basis, is equivalent to the hierarchical\napproach,\nas described by, e.g., Bungartz and Griebel [4]. However, by presenting it as\nan extension of the\ntwo-scale method, we can give an elementary treatment of its analysis\nand implementation.\nFor each method considered, we provide MATLAB code, and a comparison of accuracy and computational costs.<\/jats:p>","DOI":"10.1515\/cmam-2016-0042","type":"journal-article","created":{"date-parts":[[2017,1,10]],"date-time":"2017-01-10T10:01:27Z","timestamp":1484042487000},"page":"299-322","source":"Crossref","is-referenced-by-count":3,"title":["An Introduction to the Analysis and Implementation of Sparse Grid Finite Element Methods"],"prefix":"10.1515","volume":"17","author":[{"given":"Stephen","family":"Russell","sequence":"first","affiliation":[{"name":"School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland"}]},{"given":"Niall","family":"Madden","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland"}]}],"member":"374","published-online":{"date-parts":[[2017,1,6]]},"reference":[{"key":"2023033114222307425_j_cmam-2016-0042_ref_001_w2aab3b7e1560b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"Alberty J., Carstensen C. and Funken S. A.,\nRemarks around 50 lines of Matlab: Short finite element implementation,\nNumer. Algorithms 20 (1999), no. 2\u20133, 117\u2013137.","DOI":"10.1023\/A:1019155918070"},{"key":"2023033114222307425_j_cmam-2016-0042_ref_002_w2aab3b7e1560b1b6b1ab2b2b2Aa","doi-asserted-by":"crossref","unstructured":"Brenner S. C. and Scott R. L.,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033114222307425_j_cmam-2016-0042_ref_003_w2aab3b7e1560b1b6b1ab2b2b3Aa","doi-asserted-by":"crossref","unstructured":"Briggs W. L., Henson V. E. and McCormick S. 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