{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T22:10:48Z","timestamp":1759702248990},"reference-count":44,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,3,26]]},"abstract":"<jats:title>Abstract<\/jats:title>\n<jats:p>We consider a singularly perturbed linear reaction\u2013diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called \u2018balanced\u2019 norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction\u2013diffusion problem, and prove convergence of their computed solutions in the balanced norm.<\/jats:p>","DOI":"10.1515\/jnma-2017-0079","type":"journal-article","created":{"date-parts":[[2018,2,2]],"date-time":"2018-02-02T10:01:52Z","timestamp":1517565712000},"page":"37-55","source":"Crossref","is-referenced-by-count":8,"title":["Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction\u2013diffusion problems"],"prefix":"10.1515","volume":"27","author":[{"given":"Stephen","family":"Russell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Stynes","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"key":"ref331","volume-title":"Lecture Notes in Mathematics","volume":"Vol. 1985","year":"2010"},{"key":"ref201","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1515\/cmam-2016-0042","article-title":"An introduction to the analysis and implementation of sparse grid finite element methods","volume":"17","year":"2017","journal-title":"Comput. 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