American Institute of Mathematical Sciences

• Previous Article
Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition
• ERA Home
• This Issue
• Next Article
Recent progress on stable and finite Morse index solutions of semilinear elliptic equations
doi: 10.3934/era.2021043
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

 1 Department of Mathematics, Ajou University, Suwon 16499, Korea 2 Department of Mathematics, Myongji University, Yongin 17058, Korea

* Corresponding author

Received  September 2020 Revised  April 2021 Early access June 2021

Fund Project: Jeon was supported by NRF 2018R1D1A1A09082082 and Shin was supported by 2020 Research Fund of Myongji University

We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.

Citation: Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, doi: 10.3934/era.2021043
References:
 [1] T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64 (1995), 943-972.  doi: 10.2307/2153478.  Google Scholar [2] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar [3] F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138 (1997), 251-285.  doi: 10.1006/jcph.1997.5454.  Google Scholar [4] D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, 2$^{nd}$ edition, Cambridge University Press, 2001.  Google Scholar [5] S. C. Brenner, M. Neilan and L.-Y. Sung, Isoparametric $C^0$ interior penalty methods for plate bending problems on smooth domains, Calcolo, 50 (2013), 35-67.  doi: 10.1007/s10092-012-0057-1.  Google Scholar [6] F. Brezzi, J. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.  doi: 10.1007/BF01389710.  Google Scholar [7] F. Brezzi, L. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903.  doi: 10.1142/S0218202504003866.  Google Scholar [8] B. Camp, T. Lin, Y. Lin and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 24 (2006), 81-112.  doi: 10.1007/s10444-004-4139-8.  Google Scholar [9] B. Cockburn, D. A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 50 (2016), 635-650.  doi: 10.1051/m2an/2015051.  Google Scholar [10] A. Ern and J.-L. Guermond, Discontinuous Galerkin Methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  doi: 10.1137/050624133.  Google Scholar [11] R. P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), 457-492.  doi: 10.1006/jcph.1999.6236.  Google Scholar [12] R. P. Fedkiw, T. Aslam and S. Xu, The ghost fluid method for deflagration and detonation discontinuities, J. Comput. Phys., 154 (1999), 393-427.  doi: 10.1006/jcph.1999.6320.  Google Scholar [13] F. Fryklund, E. Lehto and A.-K. Tornberg, Partition of unity extension of functions on complex domains, J. Comput. Phys., 375 (2018), 57-79.  doi: 10.1016/j.jcp.2018.08.012.  Google Scholar [14] Y. Jeon, An immersed hybrid difference method for the elliptic interface equation, preprint. doi: 10.13140/RG.2.2.27746.58566.  Google Scholar [15] Y. Jeon, Hybrid difference methods for PDEs, J. Sci. Comput., 64 (2015), 508-521.  doi: 10.1007/s10915-014-9941-y.  Google Scholar [16] Y. Jeon, Hybrid spectral difference methods for elliptic equations on exterior domains with the discrete radial absorbing boundary condition, J. Sci. Comput., 75 (2018), 889-905.  doi: 10.1007/s10915-017-0570-0.  Google Scholar [17] Y. Jeon, E.-J. Park and D.-W. Shin, Hybrid spectral difference methods for an elliptic equation, Comput. Methods Appl. Math., 17 (2017), 253-267.  doi: 10.1515/cmam-2016-0043.  Google Scholar [18] Y. Jeon and D. Sheen, Upwind hybrid spectral difference methods for steady-state Navier–Stokes equations, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer International Publishing (eds. J. Dick, F.Y. Kuo and H. Woźniakowski), (2018), 621–644.  Google Scholar [19] L. Krivodonova and M. Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys., 211 (2006), 492-512.  doi: 10.1016/j.jcp.2005.05.029.  Google Scholar [20] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.  doi: 10.1016/0021-9991(92)90324-R.  Google Scholar [21] R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019-1044.  doi: 10.1137/0731054.  Google Scholar [22] R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), 709-735.  doi: 10.1137/S1064827595282532.  Google Scholar [23] A. N. Marques, J.-C. Nave and R. R. Rosales, High order solution of Poisson problems with piecewise constant coefficients and interface jumps, J. Comput. Phys., 335 (2017), 497-515.  doi: 10.1016/j.jcp.2017.01.029.  Google Scholar [24] A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), 285-299.  doi: 10.1137/0721021.  Google Scholar [25] A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. Statist. Comput., 6 (1985), 144-157.  doi: 10.1137/0906012.  Google Scholar [26] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar [27] C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar [28] D. Shirokoff and J.-C. Nave, A sharp-interface active penalty method for the incompressible Navier–Stokes equations, J. Sci. Comput., 62 (2015), 53-77.  doi: 10.1007/s10915-014-9849-6.  Google Scholar [29] D. B. Stein, R. D. Guy and B. Thomases, Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods, J. Comput. Phys., 304 (2016), 252-274.  doi: 10.1016/j.jcp.2015.10.023.  Google Scholar [30] Y. Wang and J. Zhang, Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation, J. Comput. Phys., 228 (2009), 137-146.  doi: 10.1016/j.jcp.2008.09.002.  Google Scholar [31] A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar [32] Y. Xie and W. Ying, A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions, J. Comput. Phys., 415 (2020), 109526, 29 pp. doi: 10.1016/j.jcp.2020.109526.  Google Scholar [33] W. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phys., 227 (2007), 1046-1074.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar [34] W. Ying and W.-C. Wang, A kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606-624.  doi: 10.1016/j.jcp.2013.06.019.  Google Scholar

show all references

References:
 [1] T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64 (1995), 943-972.  doi: 10.2307/2153478.  Google Scholar [2] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar [3] F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138 (1997), 251-285.  doi: 10.1006/jcph.1997.5454.  Google Scholar [4] D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, 2$^{nd}$ edition, Cambridge University Press, 2001.  Google Scholar [5] S. C. Brenner, M. Neilan and L.-Y. Sung, Isoparametric $C^0$ interior penalty methods for plate bending problems on smooth domains, Calcolo, 50 (2013), 35-67.  doi: 10.1007/s10092-012-0057-1.  Google Scholar [6] F. Brezzi, J. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.  doi: 10.1007/BF01389710.  Google Scholar [7] F. Brezzi, L. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903.  doi: 10.1142/S0218202504003866.  Google Scholar [8] B. Camp, T. Lin, Y. Lin and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 24 (2006), 81-112.  doi: 10.1007/s10444-004-4139-8.  Google Scholar [9] B. Cockburn, D. A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 50 (2016), 635-650.  doi: 10.1051/m2an/2015051.  Google Scholar [10] A. Ern and J.-L. Guermond, Discontinuous Galerkin Methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  doi: 10.1137/050624133.  Google Scholar [11] R. P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), 457-492.  doi: 10.1006/jcph.1999.6236.  Google Scholar [12] R. P. Fedkiw, T. Aslam and S. Xu, The ghost fluid method for deflagration and detonation discontinuities, J. Comput. Phys., 154 (1999), 393-427.  doi: 10.1006/jcph.1999.6320.  Google Scholar [13] F. Fryklund, E. Lehto and A.-K. Tornberg, Partition of unity extension of functions on complex domains, J. Comput. Phys., 375 (2018), 57-79.  doi: 10.1016/j.jcp.2018.08.012.  Google Scholar [14] Y. Jeon, An immersed hybrid difference method for the elliptic interface equation, preprint. doi: 10.13140/RG.2.2.27746.58566.  Google Scholar [15] Y. Jeon, Hybrid difference methods for PDEs, J. Sci. Comput., 64 (2015), 508-521.  doi: 10.1007/s10915-014-9941-y.  Google Scholar [16] Y. Jeon, Hybrid spectral difference methods for elliptic equations on exterior domains with the discrete radial absorbing boundary condition, J. Sci. Comput., 75 (2018), 889-905.  doi: 10.1007/s10915-017-0570-0.  Google Scholar [17] Y. Jeon, E.-J. Park and D.-W. Shin, Hybrid spectral difference methods for an elliptic equation, Comput. Methods Appl. Math., 17 (2017), 253-267.  doi: 10.1515/cmam-2016-0043.  Google Scholar [18] Y. Jeon and D. Sheen, Upwind hybrid spectral difference methods for steady-state Navier–Stokes equations, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer International Publishing (eds. J. Dick, F.Y. Kuo and H. Woźniakowski), (2018), 621–644.  Google Scholar [19] L. Krivodonova and M. Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys., 211 (2006), 492-512.  doi: 10.1016/j.jcp.2005.05.029.  Google Scholar [20] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.  doi: 10.1016/0021-9991(92)90324-R.  Google Scholar [21] R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019-1044.  doi: 10.1137/0731054.  Google Scholar [22] R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), 709-735.  doi: 10.1137/S1064827595282532.  Google Scholar [23] A. N. Marques, J.-C. Nave and R. R. Rosales, High order solution of Poisson problems with piecewise constant coefficients and interface jumps, J. Comput. Phys., 335 (2017), 497-515.  doi: 10.1016/j.jcp.2017.01.029.  Google Scholar [24] A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), 285-299.  doi: 10.1137/0721021.  Google Scholar [25] A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. Statist. Comput., 6 (1985), 144-157.  doi: 10.1137/0906012.  Google Scholar [26] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar [27] C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar [28] D. Shirokoff and J.-C. Nave, A sharp-interface active penalty method for the incompressible Navier–Stokes equations, J. Sci. Comput., 62 (2015), 53-77.  doi: 10.1007/s10915-014-9849-6.  Google Scholar [29] D. B. Stein, R. D. Guy and B. Thomases, Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods, J. Comput. Phys., 304 (2016), 252-274.  doi: 10.1016/j.jcp.2015.10.023.  Google Scholar [30] Y. Wang and J. Zhang, Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation, J. Comput. Phys., 228 (2009), 137-146.  doi: 10.1016/j.jcp.2008.09.002.  Google Scholar [31] A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar [32] Y. Xie and W. Ying, A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions, J. Comput. Phys., 415 (2020), 109526, 29 pp. doi: 10.1016/j.jcp.2020.109526.  Google Scholar [33] W. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phys., 227 (2007), 1046-1074.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar [34] W. Ying and W.-C. Wang, A kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606-624.  doi: 10.1016/j.jcp.2013.06.019.  Google Scholar
Configuration of the problem domain
One dimensional cell configuration: cell-point, intercell point
An illustration of the extended solutions $\widetilde{ u}^{\pm}(x)$ in the interface cell $I_{k-1}$: the solid lines are the original $u^{\pm}(x)$ and the dotted lines are extensions
The $P_3$-interface cell configuration. The cell interior points are Gaussian points for the interval $[\eta_{4k}, \eta_{4k+3}]$
The $Q_2^*$-grid: $|R_1| = h_1 \times k_1$, $|R_2| = h_2 \times k_1$, $|R_3| = h_1 \times k_2$
The coherent $Q_2^*$-grid (left) and $Q_3^*$-grid cells (center), and the incoherent interface cell (right). There is no rule for the choice of points $\tau_i$, and a quasi-uniformly distributed set of points along the interface will be reasonable
Numerical experiments for the 1-d Dirichlet condition on the $P_3$-grid. The first two figures represent error and convergence rate, respectively, and the right represents the scaled condition number ($(\text{Condition Number}) \times h^3$)
Numerical results for the 1-d Neumann condition on the $P_3$-grid. The scaled condition numbers are almost coincident
The three different problem domains with the $Q_3^*$-grid. Grid 1 (left, circle), Grid 2 (middle, rotated ellipse) and Grid 3 (right, five-leafed flower)
Coherence ratio for the uniform $Q_3^*$-grids with the circular interface(red), elliptic interface (blue) and flower-like interface (black). The horizontal axis represents the number of subdivisions for each axis of the computational domai
The $L_2$ errors and convergence rates for the Dirichlet condition on the circle of radius $0.3$ with the uniform $Q_3^*$-grid(Grid 1)
The $L_2$ errors and convergence rates for the Neumann condition on the circle of radius $0.3$ with the uniform $Q_3^*$-grid(Grid 1)
The $L_2$ errors and convergence rates for the Dirichlet condition on the rotated ellipse with the uniform $Q_3^*$-grid(Grid 2)
The $L_2$ errors and convergence rates for the Dirichlet condition on the five-leafed flower with the quasi-uniform $Q_3^*$-grid(Grid 3)
 [1] Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 [2] Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 [3] J. García-Melián, Julio D. Rossi, José Sabina de Lis. A convex-concave elliptic problem with a parameter on the boundary condition. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1095-1124. doi: 10.3934/dcds.2012.32.1095 [4] Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355 [5] Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 [6] Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807 [7] Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1185-1203. doi: 10.3934/dcdsb.2012.17.1185 [8] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [9] Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, , () : -. doi: 10.3934/era.2021031 [10] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 [11] Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030 [12] Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333 [13] Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31 [14] Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323 [15] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175 [16] Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure & Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785 [17] Paolo Piersanti. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021145 [18] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [19] Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371 [20] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

2020 Impact Factor: 1.833

Tools

Article outline

Figures and Tables