{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:54:02Z","timestamp":1759838042529},"reference-count":31,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"3","license":[{"start":{"date-parts":[[2017,9,1]],"date-time":"2017-09-01T00:00:00Z","timestamp":1504224000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.<\/jats:p>","DOI":"10.1515\/amcs-2017-0037","type":"journal-article","created":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T10:00:43Z","timestamp":1506333643000},"page":"527-537","source":"Crossref","is-referenced-by-count":1,"title":["Accurate gradient computations at interfaces using finite element methods"],"prefix":"10.61822","volume":"27","author":[{"given":"Fangfang","family":"Qin","sequence":"first","affiliation":[{"name":"School of Science , Nanjing University of Posts and Telecommunications , Nanjing , Jiangsu, 210023 China"}]},{"given":"Zhaohui","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics , North Carolina State University , Raleigh , NC 27695 , United States of America"}]},{"given":"Zhijie","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Resource and Environment , Wuhan University of Technology , Wuhan , 430070 China"},{"name":"China Institute of Water Resource and Hydropower Research (IWHR) , Beijing , 100038 China"}]},{"given":"Zhilin","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics , North Carolina State University , Raleigh , NC 27695 , United States of America"}]}],"member":"37438","published-online":{"date-parts":[[2017,9,23]]},"reference":[{"key":"2021040800094277988_j_amcs-2017-0037_ref_001_w2aab3b7b6b1b6b1ab1ab1Aa","unstructured":"Adams, R. and Fournier, J. (2003). Sobolev Spaces. Second Edition, Academic Press, Cambridge, MA."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_002_w2aab3b7b6b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"An, N. and Chen, H. (2014). A partially penalty immersed interface finite element method for anisotropic elliptic interface problems, Numerical Methods for Partial Differential Equations30(6): 1984\u20132028.","DOI":"10.1002\/num.21886"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_003_w2aab3b7b6b1b6b1ab1ab3Aa","unstructured":"Anitescu, C. (2017). Open source 3D Matlab isogeometric analysis code, https:\/\/sourceforge.net\/u\/cmechanicsos\/profile\/."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_004_w2aab3b7b6b1b6b1ab1ab4Aa","unstructured":"Babu\u0161ka, I. (1970). The finite element method for elliptic equations with discontinuous coefficients, Computing5(3): 207\u2013213.10.1007\/BF02248021"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_005_w2aab3b7b6b1b6b1ab1ab5Aa","unstructured":"Bramble, J. and King, J. (1996). A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics6(1): 109\u2013138.10.1007\/BF02127700"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_006_w2aab3b7b6b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"Brenner, S. and Scott, R. (2007). The Mathematical Theory of Finite Element Methods, Springer, New York, NY.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_007_w2aab3b7b6b1b6b1ab1ab7Aa","unstructured":"Cao, W., Zhang, X. and Zhang, Z. (2017). Superconvergence of immersed finite element methods for interface problems, Advances in Computational Mathematics43(4): 795\u2013821.10.1007\/s10444-016-9507-7"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_008_w2aab3b7b6b1b6b1ab1ab8Aa","unstructured":"Carstensen, C., Gallistl, D., Hellwing, F. and Weggler, L. (2014). Low-order DPG-FEM for an elliptic PDE, Computers & Mathematics with Applications68(11): 1503\u20131512.10.1016\/j.camwa.2014.09.013"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_009_w2aab3b7b6b1b6b1ab1ab9Aa","unstructured":"Chen, Z. and Zou, J. (1998). Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik79(2): 175\u2013202.10.1007\/s002110050336"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_010_w2aab3b7b6b1b6b1ab1ac10Aa","unstructured":"Chou, S. (2012). An immersed linear finite element method with interface flux capturing recovery, Discrete and Continuous Dynamical Systems B17(7): 2343\u20132357.10.3934\/dcdsb.2012.17.2343"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_011_w2aab3b7b6b1b6b1ab1ac11Aa","unstructured":"Chou, S.H., Kwak, D.Y. and Wee, K. (2010). Optimal convergence analysis of an immersed interface finite element method, Advances in Computational Mathematics33(2): 149\u2013168.10.1007\/s10444-009-9122-y"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_012_w2aab3b7b6b1b6b1ab1ac12Aa","unstructured":"Douglas Jr, J., Dupont, T. and Wheeler, M. (1974). A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems, Revue fran\u00e7aise d\u2019automatique, informatique, recherche op\u00e9rationnelle. Analyse num\u00e9rique8(2): 47\u201359."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_013_w2aab3b7b6b1b6b1ab1ac13Aa","unstructured":"Guo, H. and Yang, X. (2017). Gradient recovery for elliptic interface problem. II: Immersed finite element methods, Journal of Computational Physics338: 606\u2013619."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_014_w2aab3b7b6b1b6b1ab1ac14Aa","unstructured":"He, X., Lin, T. and Lin, Y. (2011). Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, International Journal of Numerical Analysis and Modeling8(2): 284\u2013301."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_015_w2aab3b7b6b1b6b1ab1ac15Aa","doi-asserted-by":"crossref","unstructured":"Ji, H., Chen, J. and Li, Z. (2016). A new augmented immersed finite element method without using SVD interpolations, Numerical Algorithms71(2): 395\u2013416.","DOI":"10.1007\/s11075-015-9999-0"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_016_w2aab3b7b6b1b6b1ab1ac16Aa","unstructured":"Karczewska, A., Rozmej P., Szczeci\u0144ski, M. and Boguniewicz, B. (2016). A finite element method for extended KdV equations, International Journal of Applied Mathematics and Computer Science26(3): 555\u2013567, DOI: 10.1515\/amcs-2016-0039.10.1515\/amcs-2016-0039"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_017_w2aab3b7b6b1b6b1ab1ac17Aa","doi-asserted-by":"crossref","unstructured":"Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution, Springer, New York, NY.","DOI":"10.1007\/978-1-4684-9022-0"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_018_w2aab3b7b6b1b6b1ab1ac18Aa","doi-asserted-by":"crossref","unstructured":"Kwak, D.Y., Wee, K. and Chang, K. (2010). An analysis of a broken p1 nonconforming finite element method for interface problems, SIAM Journal on Numerical Analysis48(6): 2117\u20132134.","DOI":"10.1137\/080728056"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_019_w2aab3b7b6b1b6b1ab1ac19Aa","doi-asserted-by":"crossref","unstructured":"Li, Z. (1998). The immersed interface method using a finite element formulation, Applied Numerical Mathematics27(3): 253\u2013267.","DOI":"10.1016\/S0168-9274(98)00015-4"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_020_w2aab3b7b6b1b6b1ab1ac20Aa","unstructured":"Li, Z. and Ito, K. (2006). The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, SIAM, Philadelphia, PA.10.1137\/1.9780898717464"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_021_w2aab3b7b6b1b6b1ab1ac21Aa","doi-asserted-by":"crossref","unstructured":"Li, Z., Lin, T. and Wu, X. (2003). New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik96(1): 61\u201398.","DOI":"10.1007\/s00211-003-0473-x"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_022_w2aab3b7b6b1b6b1ab1ac22Aa","doi-asserted-by":"crossref","unstructured":"Lin, T., Lin, Y. and Zhang, X. (2015). Partially penalized immersed finite element methods for elliptic interface problems, SIAM Journal on Numerical Analysis53(2): 1121\u20131144.","DOI":"10.1137\/130912700"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_023_w2aab3b7b6b1b6b1ab1ac23Aa","doi-asserted-by":"crossref","unstructured":"Lin, T. and Zhang, X. (2012). Linear and bilinear immersed finite elements for planar elasticity interface problems, Journal of Computational and Applied Mathematics236(18): 4681\u20134699.","DOI":"10.1016\/j.cam.2012.03.012"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_024_w2aab3b7b6b1b6b1ab1ac24Aa","unstructured":"Sutton, A. and Balluffi, R. (1995). Interfaces in Crystalline Materials, Clarendon Press, Oxford."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_025_w2aab3b7b6b1b6b1ab1ac25Aa","unstructured":"Tartar, L. (2007). An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, New York, NY."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_026_w2aab3b7b6b1b6b1ab1ac26Aa","unstructured":"Wahlbin, L. (1995). Superconvergence in Galerkin Finite Element Methods, Springer, New York, NY.10.1007\/BFb0096835"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_027_w2aab3b7b6b1b6b1ab1ac27Aa","unstructured":"Wheeler, M. (1974). A Galerkin procedure for estimating the flux for two-point boundary value problems, SIAM Journal on Numerical Analysis11(4): 764\u2013768.10.1137\/0711062"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_028_w2aab3b7b6b1b6b1ab1ac28Aa","unstructured":"Xu, J. (1982). Error estimates of the finite element method for the 2nd order elliptic equations with discontinuous coefficients, Journal of Xiangtan University1(1): 1\u20135."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_029_w2aab3b7b6b1b6b1ab1ac29Aa","unstructured":"Yang, X., Li, B. and Li, Z. (2002). The immersed interface method for elasticity problems with interfaces, Dynamics of Continuous, Discrete and Impulsive Systems10(5): 783\u2013808."},{"key":"2021040800094277988_j_amcs-2017-0037_ref_030_w2aab3b7b6b1b6b1ab1ac30Aa","unstructured":"Zhang, Z. and Naga, A. (2005). A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing26(4): 1192\u20131213.10.1137\/S1064827503402837"},{"key":"2021040800094277988_j_amcs-2017-0037_ref_031_w2aab3b7b6b1b6b1ab1ac31Aa","unstructured":"Zienkiewicz, O. and Taylor, R. (2000). The Finite Element Method: Solid Mechanics, Butterworth-Heinemann, Oxford."}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/27\/3\/article-p527.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/amcs-2017-0037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T22:57:25Z","timestamp":1715813845000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/amcs-2017-0037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,1]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9,23]]},"published-print":{"date-parts":[[2017,9,1]]}},"alternative-id":["10.1515\/amcs-2017-0037"],"URL":"https:\/\/doi.org\/10.1515\/amcs-2017-0037","relation":{},"ISSN":["2083-8492"],"issn-type":[{"value":"2083-8492","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,9,1]]}}}