Accurate gradient computations at interfaces using finite element methods

Adams, R. and Fournier, J. (2003). Sobolev Spaces. Second Edition, Academic Press, Cambridge, MA.Search in Google Scholar

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): 1984–2028.10.1002/num.21886Search in Google Scholar

Anitescu, C. (2017). Open source 3D Matlab isogeometric analysis code, https://sourceforge.net/u/cmechanicsos/profile/.Search in Google Scholar

Babuška, I. (1970). The finite element method for elliptic equations with discontinuous coefficients, Computing5(3): 207–213.10.1007/BF02248021Open DOISearch in Google Scholar

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): 109–138.10.1007/BF02127700Open DOISearch in Google Scholar

Brenner, S. and Scott, R. (2007). The Mathematical Theory of Finite Element Methods, Springer, New York, NY.10.1007/978-0-387-75934-0Search in Google Scholar

Cao, W., Zhang, X. and Zhang, Z. (2017). Superconvergence of immersed finite element methods for interface problems, Advances in Computational Mathematics43(4): 795–821.10.1007/s10444-016-9507-7Open DOISearch in Google Scholar

Carstensen, C., Gallistl, D., Hellwing, F. and Weggler, L. (2014). Low-order DPG-FEM for an elliptic PDE, Computers & Mathematics with Applications68(11): 1503–1512.10.1016/j.camwa.2014.09.013Open DOISearch in Google Scholar

Chen, Z. and Zou, J. (1998). Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik79(2): 175–202.10.1007/s002110050336Open DOISearch in Google Scholar

Chou, S. (2012). An immersed linear finite element method with interface flux capturing recovery, Discrete and Continuous Dynamical Systems B17(7): 2343–2357.10.3934/dcdsb.2012.17.2343Open DOISearch in Google Scholar

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): 149–168.10.1007/s10444-009-9122-yOpen DOISearch in Google Scholar

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 française d’automatique, informatique, recherche opérationnelle. Analyse numérique8(2): 47–59.Search in Google Scholar

Guo, H. and Yang, X. (2017). Gradient recovery for elliptic interface problem. II: Immersed finite element methods, Journal of Computational Physics338: 606–619.Search in Google Scholar

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): 284–301.Search in Google Scholar

Ji, H., Chen, J. and Li, Z. (2016). A new augmented immersed finite element method without using SVD interpolations, Numerical Algorithms71(2): 395–416.10.1007/s11075-015-9999-0Search in Google Scholar

Karczewska, A., Rozmej P., Szczeciński, M. and Boguniewicz, B. (2016). A finite element method for extended KdV equations, International Journal of Applied Mathematics and Computer Science26(3): 555–567, DOI: 10.1515/amcs-2016-0039.10.1515/amcs-2016-0039Open DOISearch in Google Scholar

Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution, Springer, New York, NY.10.1007/978-1-4684-9022-0Search in Google Scholar

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): 2117–2134.10.1137/080728056Search in Google Scholar

Li, Z. (1998). The immersed interface method using a finite element formulation, Applied Numerical Mathematics27(3): 253–267.10.1016/S0168-9274(98)00015-4Search in Google Scholar

Li, Z. and Ito, K. (2006). The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, SIAM, Philadelphia, PA.10.1137/1.9780898717464Open DOISearch in Google Scholar

Li, Z., Lin, T. and Wu, X. (2003). New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik96(1): 61–98.10.1007/s00211-003-0473-xSearch in Google Scholar

Lin, T., Lin, Y. and Zhang, X. (2015). Partially penalized immersed finite element methods for elliptic interface problems, SIAM Journal on Numerical Analysis53(2): 1121–1144.10.1137/130912700Search in Google Scholar

Lin, T. and Zhang, X. (2012). Linear and bilinear immersed finite elements for planar elasticity interface problems, Journal of Computational and Applied Mathematics236(18): 4681–4699.10.1016/j.cam.2012.03.012Search in Google Scholar

Sutton, A. and Balluffi, R. (1995). Interfaces in Crystalline Materials, Clarendon Press, Oxford.Search in Google Scholar

Tartar, L. (2007). An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, New York, NY.Search in Google Scholar

Wahlbin, L. (1995). Superconvergence in Galerkin Finite Element Methods, Springer, New York, NY.10.1007/BFb0096835Open DOISearch in Google Scholar

Wheeler, M. (1974). A Galerkin procedure for estimating the flux for two-point boundary value problems, SIAM Journal on Numerical Analysis11(4): 764–768.10.1137/0711062Open DOISearch in Google Scholar

Xu, J. (1982). Error estimates of the finite element method for the 2nd order elliptic equations with discontinuous coefficients, Journal of Xiangtan University1(1): 1–5.Search in Google Scholar

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): 783–808.Search in Google Scholar

Zhang, Z. and Naga, A. (2005). A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing26(4): 1192–1213.10.1137/S1064827503402837Open DOISearch in Google Scholar

Zienkiewicz, O. and Taylor, R. (2000). The Finite Element Method: Solid Mechanics, Butterworth-Heinemann, Oxford.Search in Google Scholar