## Stynes, Martin

Compute Distance To:
 Author ID: stynes.martin Published as: Stynes, Martin; Stynes, M.
 Documents Indexed: 177 Publications since 1977, including 5 Books 3 Contributions as Editor Co-Authors: 57 Co-Authors with 158 Joint Publications 1,397 Co-Co-Authors
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### Co-Authors

 20 single-authored 31 O’Riordan, Eugene 17 Gracia, José Luis 13 Kellogg, Bruce 11 Huang, Chaobao 10 Kopteva, Natalia Viktorovna 10 Tobiska, Lutz 9 Roos, Hans-Görg 7 Chen, Hu 7 Madden, Niall 6 Linß, Torsten 6 Meng, XiangYun 5 Sun, Guangfu 4 Guo, Wen 4 Viscor, Martin 3 Huang, Can 3 Le, Kim-Ngan 3 Liu, Fang 3 Morrison, Clarence Clapp 3 Naughton, Aidan 3 Zhou, Aihui 3 Zhou, Yongtao 2 Franz, Sebastian 2 Hegarty, Alan F. 2 Liang, Hui 2 Lin, Runchang 2 Liu, Xiaohui 2 Morton, Keith William 2 Natividad, Maria Caridad 2 Shishkin, Grigoriĭ Ivanovich 2 Shishkina, Lidia P. 2 Stynes, Jeanne 2 Zhang, Jin 1 Adam, Dirk 1 An, Na 1 Angermann, Lutz 1 Clavero, Carmelo 1 Diethelm, Kai 1 Felgenhauer, Andreas 1 Franca, Leopoldo Luis Cabo Penna 1 Garrappa, Roberto 1 Giusti, Andrea 1 Grossmann, Christian 1 Holland, Finbarr 1 Huang, Zhongyi 1 Liu, Xiaowei 1 Ma, Guanglong 1 McLean, William 1 Nesliturk, Ali I. 1 Ng-Stynes, Mary Jeanne 1 Nhan, Thái Anh 1 Russell, Stephen J. 1 Shen, Jinye 1 Stynes, David 1 Süli, Endre E. 1 Sun, Zhizhong 1 Vulanović, Relja 1 Zhang, Zhimin
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### Serials

 12 IMA Journal of Numerical Analysis 12 Mathematics of Computation 10 Journal of Computational and Applied Mathematics 10 SIAM Journal on Numerical Analysis 9 Numerische Mathematik 9 Applied Numerical Mathematics 9 Journal of Scientific Computing 9 Computational Methods in Applied Mathematics 7 Computers & Mathematics with Applications 6 BIT 6 Numerical Algorithms 6 Advances in Computational Mathematics 4 Applicable Analysis 4 Fractional Calculus & Applied Analysis 3 Computer Methods in Applied Mechanics and Engineering 3 Journal of Mathematical Analysis and Applications 3 Applied Mathematics Letters 2 Computing 2 Journal of Differential Equations 2 Irish Mathematical Society Newsletter 2 Numerical Methods for Partial Differential Equations 2 Applications of Mathematics 2 Mathematical Modelling and Analysis 2 Springer Series in Computational Mathematics 2 Lecture Notes in Computational Science and Engineering 2 East Asian Journal on Applied Mathematics 1 Journal of Computational Physics 1 Mathematics Magazine 1 Calcolo 1 Chinese Journal of Mathematics 1 Rendiconti del Seminario Matemàtico e Fisico di Milano 1 Zeitschrift für Analysis und ihre Anwendungen 1 RAIRO. Modélisation Mathématique et Analyse Numérique 1 CWI Quarterly 1 Mathematica Bohemica 1 Proceedings of the Royal Irish Academy. Section A, Mathematical and Physical Sciences 1 Irish Mathematical Society Bulletin 1 East-West Journal of Numerical Mathematics 1 Journal of Numerical Mathematics 1 Communications on Pure and Applied Analysis 1 Acta Numerica 1 International Journal of Numerical Analysis and Modeling 1 Graduate Studies in Mathematics 1 Numerical Mathematics: Theory, Methods and Applications 1 International Journal of Computing Science and Mathematics 1 Universitext
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### Fields

 158 Numerical analysis (65-XX) 93 Partial differential equations (35-XX) 55 Ordinary differential equations (34-XX) 13 Real functions (26-XX) 9 Algebraic topology (55-XX) 5 Integral equations (45-XX) 5 Fluid mechanics (76-XX) 3 General and overarching topics; collections (00-XX) 3 Approximations and expansions (41-XX) 2 Operator theory (47-XX) 2 Geometry (51-XX) 2 General topology (54-XX) 2 Manifolds and cell complexes (57-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Mechanics of deformable solids (74-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

### Citations contained in zbMATH Open

146 Publications have been cited 3,223 times in 1,793 Documents Cited by Year
Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. 2nd ed. Zbl 1155.65087
Roos, Hans-Görg; Stynes, Martin; Tobiska, Lutz
2008
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. Zbl 1362.65089
Stynes, Martin; O’Riordan, Eugene; Gracia, José Luis
2017
Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. Zbl 0844.65075
Roos, Hans-Görg; Stynes, Martin; Tobiska, Lutz
1996
Stynes, Martin
2005
The midpoint upwind scheme. Zbl 0877.65055
Stynes, Martin; Roos, Hans-Görg
1997
Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem. Zbl 1200.35046
Linß, Torsten; Stynes, Martin
2001
A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. Zbl 0917.65088
Stynes, Martin; O’Riordan, Eugene
1997
Too much regularity may force too much uniqueness. Zbl 1353.35306
Stynes, Martin
2016
A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. Zbl 1048.65076
2003
The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy. Zbl 1055.65121
Stynes, Martin; Tobiska, Lutz
2003
A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem. Zbl 1012.65076
Kopteva, Natalia; Stynes, Martin
2001
A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. Zbl 1339.65097
Stynes, Martin; Gracia, José Luis
2015
A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Zbl 0733.65063
O’Riordan, Eugene; Stynes, Martin
1991
On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Zbl 0934.65127
Franca, L. P.; Nesliturk, A.; Stynes, M.
1998
Corner singularities and boundary layers in a simple convection-diffusion problem. Zbl 1159.35309
Kellogg, R. Bruce; Stynes, Martin
2005
Numerical treatment of partial differential equations. Revised translation of the 3rd German edition of ‘Numerische Behandlung partieller Differentialgleichungen’ by Martin Stynes. Zbl 1180.65147
Grossmann, Christian; Roos, Hans-Görg; Stynes, Martin
2007
Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. Zbl 1419.65010
Chen, Hu; Stynes, Martin
2019
A balanced finite element method for singularly perturbed reaction-diffusion problems. Zbl 1260.65103
Lin, Runchang; Stynes, Martin
2012
Finite-element methods for singularly perturbed high-order elliptic two- point boundary value problems. I: Reaction-diffusion-type problems. Zbl 0814.65083
Sun, Guangfu; Stynes, Martin
1995
A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem. Zbl 0625.65073
O’Riordan, Eugene; Stynes, Martin
1986
Numerical methods on Shishkin meshes for linear convection-diffusion problems. Zbl 0988.76062
Linß, Torsten; Stynes, Martin
2001
Collocation methods for general Caputo two-point boundary value problems. Zbl 1398.65180
Liang, Hui; Stynes, Martin
2018
A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Zbl 0916.65108
Stynes, Martin; Tobiska, Lutz
1998
Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem. Zbl 1381.65057
Kopteva, Natalia; Stynes, Martin
2017
Richardson extrapolation for a convection–diffusion problem using a Shishkin mesh. Zbl 1019.65053
2003
Numerical methods for time-dependent convection-diffusion equations. Zbl 0634.65108
Ng-Stynes, Mary Jeanne; O’Riordan, Eugene; Stynes, Martin
1988
Using rectangularqpelements in the SDFEM for a convection-diffusion problem with a boundary layer. Zbl 1154.65083
Stynes, Martin; Tobiska, Lutz
2008
An efficient collocation method for a Caputo two-point boundary value problem. Zbl 1337.65088
Kopteva, Natalia; Stynes, Martin
2015
A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. Zbl 1188.65153
Liu, Fang; Madden, Niall; Stynes, Martin; Zhou, Aihui
2009
Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem. Zbl 1201.35094
Kellogg, R. Bruce; Stynes, Martin
2007
Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points. Zbl 0671.65083
Stynes, Martin; O’Riordan, Eugene
1989
A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems. Zbl 0939.65117
Linß, Torsten; Stynes, Martin
1999
Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. Zbl 1295.65081
Gracia, J. L.; Stynes, M.
2015
Optimal $$L^\infty (L^2)$$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem. Zbl 1433.65212
Huang, Chaobao; Stynes, Martin; An, Na
2018
The SDFEM on Shishkin meshes for linear convection-diffusion problems. Zbl 0969.65106
Linß, Torsten; Stynes, Martin
2001
Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem. Zbl 1382.65275
Gracia, José Luis; O’Riordan, Eugene; Stynes, Martin
2018
A finite element method for a singularly perturbed boundary value problem. Zbl 0583.65054
Stynes, Martin; O’Riordan, Eugene
1986
Finite-element methods for singularly perturbed high-order elliptic two- point boundary value problems. II: Convection-diffusion-type problems. Zbl 0824.65077
Sun, Guangfu; Stynes, Martin
1995
A uniformly convergent discretization method for a fourth order singular perturbation problem. Zbl 0747.65066
Roos, Hans-Görg; Stynes, Martin
1991
A comparison of uniformly convergent difference schemes for two- dimensional convection diffusion problems. Zbl 0769.65071
Hegarty, Alan F.; O’Riordan, Eugene; Stynes, Martin
1993
Grid equidistribution for reaction-diffusion problems in one dimension. Zbl 1089.65077
Kopteva, Natalia; Madden, Niall; Stynes, Martin
2005
An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem. Zbl 0824.65078
Sun, Guangfu; Stynes, Martin
1995
A uniformly accurate finite element method for a singular perturbation problem in conservative form. Zbl 0595.65091
Stynes, Martin; O’Riordan, Eugene
1986
A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Zbl 1198.65209
Kellogg, R. Bruce; Linss, Torsten; Stynes, Martin
2008
Numerical solution of systems of singularly perturbed differential equations. Zbl 1193.65189
Linss, T.; Stynes, M.
2009
Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection-diffusion problem. Zbl 0872.65089
Guo, Wen; Stynes, Martin
1997
A fitted scheme for a Caputo initial-boundary value problem. Zbl 1397.65160
Gracia, J. L.; O’Riordan, Eugene; Stynes, M.
2018
Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems. Zbl 1167.65044
O’Riordan, Eugene; Stynes, Martin
2009
Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions. Zbl 1069.65087
Kopteva, Natalia; Stynes, Martin
2004
Analysis of a cell-vertex finite volume method for convection-diffusion problems. Zbl 0885.65121
Morton, K. W.; Stynes, Martin; Süli, Endre
1997
Approximation of derivatives in a convection-diffusion two-point boundary value problem. Zbl 0987.65069
Kopteva, Natalia; Stynes, Martin
2001
A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems. Zbl 1225.65084
Clavero, C.; Gracia, J. L.; Stynes, M.
2011
The streamline-diffusion method for nonconforming $$Q_1^{\text{rot}}$$ elements on rectangular tensor-product meshes. Zbl 0980.65125
Stynes, Martin; Tobiska, Lutz
2001
Collocation methods for general Riemann-Liouville two-point boundary value problems. Zbl 1415.65168
Liang, Hui; Stynes, Martin
2019
An analysis of a superconvergence result for a singularly perturbed boundary value problem. Zbl 0612.65043
O’Riordan, Eugene; Stynes, Martin
1986
A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions. Zbl 1133.65088
Kellogg, R. B.; Madden, Niall; Stynes, Martin
2008
On faster convergence of the bisection method for all triangles. Zbl 0463.65005
Stynes, Martin
1980
A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions. Zbl 0858.65081
Sun, Guangfu; Stynes, Martin
1996
Finite element methods for convection-diffusion problems using exponential splines on triangles. Zbl 0907.65110
Sacco, R.; Stynes, M.
1998
Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. Zbl 1447.65078
Huang, Chaobao; Stynes, Martin
2020
Some open questions in the numerical analysis of singularly perturbed differential equations. Zbl 1329.65278
Roos, Hans-Görg; Stynes, Martin
2015
A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Zbl 1361.65102
Huang, Can; Stynes, Martin
2016
An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. Zbl 0718.65062
Stynes, Martin; O’Riordan, Eugene
1991
Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem. Zbl 1232.65109
Kopteva, Natalia; Stynes, Martin
2011
Finite element methods on piecewise equidistant meshes for interior turning point problems. Zbl 0811.65059
Sun, Guangfu; Stynes, Martin
1994
Necessary $$L^ 2$$-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems. Zbl 0822.65077
Stynes, M.; Tobiska, L.
1995
An analysis of the Grünwald-Letnikov scheme for initial-value problems with weakly singular solutions. Zbl 1416.65257
Chen, Hu; Holland, Finbarr; Stynes, Martin
2019
Why fractional derivatives with nonsingular kernels should not be used. Zbl 1474.26020
Diethelm, Kai; Garrappa, Roberto; Giusti, Andrea; Stynes, Martin
2020
The Green’s function and a maximum principle for a Caputo two-point boundary value problem with a convection term. Zbl 1387.34008
Meng, Xiangyun; Stynes, Martin
2018
A simplification of Stenger’s topological degree formula. Zbl 0387.65035
Stynes, Martin
1979
Optimal spatial $$H^1$$-norm analysis of a finite element method for a time-fractional diffusion equation. Zbl 1428.65042
Huang, Chaobao; Stynes, Martin
2020
Boundary layers in a two-point boundary value problem with a Caputo fractional derivative. Zbl 1318.34010
Stynes, Martin; Gracia, José Luis
2015
On the construction of sufficient refinements for computation of topological degree. Zbl 0451.55001
Stynes, Martin
1981
Fractional-order derivatives defined by continuous kernels are too restrictive. Zbl 1401.26017
Stynes, Martin
2018
A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition. Zbl 1404.65171
Huang, Chaobao; Stynes, Martin
2019
$$L^ 1$$ and $$L^{\infty}$$ uniform convergence of a difference scheme for a semilinear singular perturbation problem. Zbl 0595.65092
Stynes, Martin; O’Riordan, Eugene
1987
Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities. Zbl 1244.65173
Franz, Sebastian; Kellogg, R. Bruce; Stynes, Martin
2012
An algorithm for numerical calculation of topological degree. Zbl 0407.65019
Stynes, Martin
1979
Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers. Zbl 0986.65109
Stynes, M.; Tobiska, L.
2001
Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Zbl 1451.65146
Huang, Chaobao; Liu, Xiaohui; Meng, Xiangyun; Stynes, Martin
2020
Optimal uniform-convergence results for convection-diffusion problems in one dimension using preconditioning. Zbl 1444.65041
Nhan, Thái Anh; Stynes, Martin; Vulanović, Relja
2018
Finite element analysis of exponentially fitted lumped schemes for time- dependent convection-diffusion problems. Zbl 0791.65076
Guo, Wen; Stynes, Martin
1993
Optimal $$H^1$$ spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation. Zbl 1454.65118
Huang, Chaobao; Stynes, Martin
2020
Convection-diffusion problems. An introduction to their analysis and numerical solution. Zbl 1422.65005
Stynes, Martin; Stynes, David
2018
Supercloseness of continuous interior penalty method for convection-diffusion problems with characteristic layers. Zbl 1439.65204
Zhang, Jin; Stynes, Martin
2017
On faster convergence of the bisection method for certain triangles. Zbl 0405.65010
Stynes, Martin
1979
A parameter-uniform finite difference method for a coupled system of convection-diffusion two-point boundary value problems. Zbl 1174.65441
O’Riordan, Eugene; Stynes, Jeanne; Stynes, Martin
2008
Preprocessing schemes for fractional-derivative problems to improve their convergence rates. Zbl 1376.65120
Stynes, Martin; Gracia, José Luis
2017
A new analysis of a numerical method for the time-fractional Fokker-Planck equation with general forcing. Zbl 1455.65170
Huang, Can; Le, Kim Ngan; Stynes, Martin
2020
An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem. Zbl 0685.65075
Stynes, Martin
1989
Layers and corner singularities in singularly perturbed elliptic problems. Zbl 1148.65089
Kellogg, R. Bruce; Stynes, Martin
2008
Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions. Zbl 1134.65356
Stynes, M.; Kopteva, N.
2006
An analysis of some exponentially fitted finite element methods for singularly perturbed elliptic problems. Zbl 0769.65082
O’Riordan, Eugene; Stynes, Martin
1991
A nonconforming finite element method for a singularly perturbed boundary value problem. Zbl 0813.65105
Adam, D.; Felgenhauer, A.; Roos, H.-G.; Stynes, M.
1995
Linear enhancements of the streamline diffusion method for convection-diffusion problems. Zbl 0870.65100
1996
Optimal approximability of solutions of singularly perturbed two-point boundary value problems. Zbl 0884.41017
Kellogg, R. Bruce; Stynes, Martin
1997
Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann-Liouville-Caputo fractional derivative. Zbl 1442.65130
Gracia, José Luis; O’Riordan, Eugene; Stynes, Martin
2020
Supercloseness of edge stabilization on Shishkin rectangular meshes for convection-diffusion problems with exponential layers. Zbl 1477.65229
Liu, Xiaowei; Stynes, Martin; Zhang, Jin
2018
A balanced finite element method for a system of singularly perturbed reaction-diffusion two-point boundary value problems. Zbl 1333.65084
Lin, Runchang; Stynes, Martin
2015
Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing. Zbl 1481.35382
Le, Kim-Ngan; McLean, William; Stynes, Martin
2019
Using complete monotonicity to deduce local error estimates for discretisations of a multi-term time-fractional diffusion equation. Zbl 1484.65175
Chen, Hu; Stynes, Martin
2022
Blow-up of error estimates in time-fractional initial-boundary value problems. Zbl 07528268
Chen, Hu; Stynes, Martin
2021
$$\alpha$$-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Zbl 1481.65184
Huang, Chaobao; Stynes, Martin
2021
An $$\alpha$$-robust finite element method for a multi-term time-fractional diffusion problem. Zbl 1457.65113
Huang, Chaobao; Stynes, Martin; Chen, Hu
2021
Block boundary value methods for linear weakly singular Volterra integro-differential equations. Zbl 1472.65170
Zhou, Yongtao; Stynes, Martin
2021
A weighted and balanced FEM for singularly perturbed reaction-diffusion problems. Zbl 1477.65231
2021
Two finite difference schemes for multi-dimensional fractional wave equations with weakly singular solutions. Zbl 1476.65191
Shen, Jinye; Stynes, Martin; Sun, Zhi-Zhong
2021
Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. Zbl 1447.65078
Huang, Chaobao; Stynes, Martin
2020
Why fractional derivatives with nonsingular kernels should not be used. Zbl 1474.26020
Diethelm, Kai; Garrappa, Roberto; Giusti, Andrea; Stynes, Martin
2020
Optimal spatial $$H^1$$-norm analysis of a finite element method for a time-fractional diffusion equation. Zbl 1428.65042
Huang, Chaobao; Stynes, Martin
2020
Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Zbl 1451.65146
Huang, Chaobao; Liu, Xiaohui; Meng, Xiangyun; Stynes, Martin
2020
Optimal $$H^1$$ spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation. Zbl 1454.65118
Huang, Chaobao; Stynes, Martin
2020
A new analysis of a numerical method for the time-fractional Fokker-Planck equation with general forcing. Zbl 1455.65170
Huang, Can; Le, Kim Ngan; Stynes, Martin
2020
Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann-Liouville-Caputo fractional derivative. Zbl 1442.65130
Gracia, José Luis; O&rsquo;Riordan, Eugene; Stynes, Martin
2020
Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation. Zbl 1437.65191
Huang, Chaobao; Stynes, Martin
2020
A discrete comparison principle for the time-fractional diffusion equation. Zbl 1447.65018
Chen, Hu; Stynes, Martin
2020
An alternative finite difference stability analysis for a multiterm time-fractional initial-boundary value problem. Zbl 1468.65098
Liu, Xiaohui; Stynes, Martin
2020
Barrier function local and global analysis of an $$L1$$ finite element method for a multiterm time-fractional initial-boundary value problem. Zbl 1447.65086
Meng, Xiangyun; Stynes, Martin
2020
A direct discontinuous Galerkin finite element method for convection-dominated two-point boundary value problems. Zbl 1440.65220
Ma, Guanglong; Stynes, Martin
2020
Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. Zbl 1419.65010
Chen, Hu; Stynes, Martin
2019
Collocation methods for general Riemann-Liouville two-point boundary value problems. Zbl 1415.65168
Liang, Hui; Stynes, Martin
2019
An analysis of the Grünwald-Letnikov scheme for initial-value problems with weakly singular solutions. Zbl 1416.65257
Chen, Hu; Holland, Finbarr; Stynes, Martin
2019
A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition. Zbl 1404.65171
Huang, Chaobao; Stynes, Martin
2019
Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing. Zbl 1481.35382
Le, Kim-Ngan; McLean, William; Stynes, Martin
2019
Superconvergence of the direct discontinuous Galerkin method for a time-fractional initial-boundary value problem. Zbl 1431.65216
Huang, Chaobao; Stynes, Martin
2019
Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions. Zbl 1415.65261
Meng, Xiangyun; Stynes, Martin
2019
Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction-diffusion problems. Zbl 1416.65464
Russell, Stephen; Stynes, Martin
2019
Green’s functions, positive solutions, and a Lyapunov inequality for a Caputo fractional-derivative boundary value problem. Zbl 1426.34017
Meng, Xiangyun; Stynes, Martin
2019
A high order method on graded meshes for a time-fractional diffusion problem. Zbl 1434.65109
Chen, Hu; Stynes, Martin
2019
A collocation method for a two-point boundary value problem with a Riemann-Liouville-Caputo fractional derivative. Zbl 1469.65126
Gracia, José Luis; O&rsquo;Riordan, Eugene; Stynes, Martin
2019
Collocation methods for general Caputo two-point boundary value problems. Zbl 1398.65180
Liang, Hui; Stynes, Martin
2018
Optimal $$L^\infty (L^2)$$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem. Zbl 1433.65212
Huang, Chaobao; Stynes, Martin; An, Na
2018
Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem. Zbl 1382.65275
Gracia, José Luis; O&rsquo;Riordan, Eugene; Stynes, Martin
2018
A fitted scheme for a Caputo initial-boundary value problem. Zbl 1397.65160
Gracia, J. L.; O&rsquo;Riordan, Eugene; Stynes, M.
2018
The Green’s function and a maximum principle for a Caputo two-point boundary value problem with a convection term. Zbl 1387.34008
Meng, Xiangyun; Stynes, Martin
2018
Fractional-order derivatives defined by continuous kernels are too restrictive. Zbl 1401.26017
Stynes, Martin
2018
Optimal uniform-convergence results for convection-diffusion problems in one dimension using preconditioning. Zbl 1444.65041
Nhan, Thái Anh; Stynes, Martin; Vulanović, Relja
2018
Convection-diffusion problems. An introduction to their analysis and numerical solution. Zbl 1422.65005
Stynes, Martin; Stynes, David
2018
Supercloseness of edge stabilization on Shishkin rectangular meshes for convection-diffusion problems with exponential layers. Zbl 1477.65229
Liu, Xiaowei; Stynes, Martin; Zhang, Jin
2018
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. Zbl 1362.65089
Stynes, Martin; O&rsquo;Riordan, Eugene; Gracia, José Luis
2017
Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem. Zbl 1381.65057
Kopteva, Natalia; Stynes, Martin
2017
Supercloseness of continuous interior penalty method for convection-diffusion problems with characteristic layers. Zbl 1439.65204
Zhang, Jin; Stynes, Martin
2017
Preprocessing schemes for fractional-derivative problems to improve their convergence rates. Zbl 1376.65120
Stynes, Martin; Gracia, José Luis
2017
Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. Zbl 1433.65353
Huang, Can; Stynes, Martin
2017
Convergence outside the initial layer for a numerical method for the time-fractional heat equation. Zbl 1368.65136
Gracia, José Luis; O&rsquo;Riordan, Eugene; Stynes, Martin
2017
Too much regularity may force too much uniqueness. Zbl 1353.35306
Stynes, Martin
2016
A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Zbl 1361.65102
Huang, Can; Stynes, Martin
2016
A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. Zbl 1339.65097
Stynes, Martin; Gracia, José Luis
2015
An efficient collocation method for a Caputo two-point boundary value problem. Zbl 1337.65088
Kopteva, Natalia; Stynes, Martin
2015
Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. Zbl 1295.65081
Gracia, J. L.; Stynes, M.
2015
Some open questions in the numerical analysis of singularly perturbed differential equations. Zbl 1329.65278
Roos, Hans-Görg; Stynes, Martin
2015
Boundary layers in a two-point boundary value problem with a Caputo fractional derivative. Zbl 1318.34010
Stynes, Martin; Gracia, José Luis
2015
A balanced finite element method for a system of singularly perturbed reaction-diffusion two-point boundary value problems. Zbl 1333.65084
Lin, Runchang; Stynes, Martin
2015
Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems. Zbl 1308.65126
Gracia, José Luis; Stynes, Martin
2015
Blow-up of solutions and interior layers in a Caputo two-point boundary value problem. Zbl 1427.65123
Stynes, Martin; Gracia, José Luis
2015
Boundary layers in a Riemann-Liouville fractional derivative two-point boundary value problem. Zbl 1427.65119
Gracia, José Luis; Stynes, Martin
2015
Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements. Zbl 1300.65079
Stynes, Martin
2014
A robust finite difference method for a singularly perturbed degenerate parabolic problem. II. Zbl 1270.65050
Viscor, Martin; Stynes, Martin
2013
A balanced finite element method for singularly perturbed reaction-diffusion problems. Zbl 1260.65103
Lin, Runchang; Stynes, Martin
2012
Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities. Zbl 1244.65173
Franz, Sebastian; Kellogg, R. Bruce; Stynes, Martin
2012
A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems. Zbl 1225.65084
Clavero, C.; Gracia, J. L.; Stynes, M.
2011
Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem. Zbl 1232.65109
Kopteva, Natalia; Stynes, Martin
2011
Postprocessed two-scale finite element discretizations. I. Zbl 1241.65101
Liu, Fang; Stynes, Martin; Zhou, Aihui
2011
A robust finite difference method for a singularly perturbed degenerate parabolic problems. I. Zbl 1198.65175
Viscor, Martin; Stynes, Martin
2010
A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. Zbl 1188.65153
Liu, Fang; Madden, Niall; Stynes, Martin; Zhou, Aihui
2009
Numerical solution of systems of singularly perturbed differential equations. Zbl 1193.65189
Linss, T.; Stynes, M.
2009
Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems. Zbl 1167.65044
O&rsquo;Riordan, Eugene; Stynes, Martin
2009
Regularity and derivative bounds for a convection-diffusion problem with a Neumann outflow condition. Zbl 1181.35054
Naughton, Aidan; Kellogg, R. Bruce; Stynes, Martin
2009
The combination technique for a two-dimensional convection-diffusion problem with exponential layers. Zbl 1212.65443
Franz, Sebastian; Liu, Fang; Roos, Hans-Görg; Stynes, Martin; Zhou, Aihui
2009
An iterative numerical algorithm for a strongly coupled system of singularly perturbed convection-diffusion problems. Zbl 1233.65079
O&rsquo;Riordan, E.; Stynes, J.; Stynes, M.
2009
Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. 2nd ed. Zbl 1155.65087
Roos, Hans-Görg; Stynes, Martin; Tobiska, Lutz
2008
Using rectangularqpelements in the SDFEM for a convection-diffusion problem with a boundary layer. Zbl 1154.65083
Stynes, Martin; Tobiska, Lutz
2008
A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Zbl 1198.65209
Kellogg, R. Bruce; Linss, Torsten; Stynes, Martin
2008
A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions. Zbl 1133.65088
Kellogg, R. B.; Madden, Niall; Stynes, Martin
2008
A parameter-uniform finite difference method for a coupled system of convection-diffusion two-point boundary value problems. Zbl 1174.65441
O&rsquo;Riordan, Eugene; Stynes, Jeanne; Stynes, Martin
2008
Layers and corner singularities in singularly perturbed elliptic problems. Zbl 1148.65089
Kellogg, R. Bruce; Stynes, Martin
2008
Numerical treatment of partial differential equations. Revised translation of the 3rd German edition of ‘Numerische Behandlung partieller Differentialgleichungen’ by Martin Stynes. Zbl 1180.65147
Grossmann, Christian; Roos, Hans-Görg; Stynes, Martin
2007
Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem. Zbl 1201.35094
Kellogg, R. Bruce; Stynes, Martin
2007
Convection-diffusion-reaction problems, SDFEM/SUPG and a priori meshes. Zbl 1185.65218
Stynes, Martin
2007
Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions. Zbl 1134.65356
Stynes, M.; Kopteva, N.
2006
A singularly perturbed convection-diffusion problem in a half-plane. Zbl 1156.35349
Kellogg, R. Bruce; Stynes, Martin
2006
Stynes, Martin
2005
Corner singularities and boundary layers in a simple convection-diffusion problem. Zbl 1159.35309
Kellogg, R. Bruce; Stynes, Martin
2005
Grid equidistribution for reaction-diffusion problems in one dimension. Zbl 1089.65077
Kopteva, Natalia; Madden, Niall; Stynes, Martin
2005
Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions. Zbl 1069.65087
Kopteva, Natalia; Stynes, Martin
2004
A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. Zbl 1048.65076
2003
The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy. Zbl 1055.65121
Stynes, Martin; Tobiska, Lutz
2003
Richardson extrapolation for a convection–diffusion problem using a Shishkin mesh. Zbl 1019.65053
2003
A jejune heuristic mesh theorem. Zbl 1040.65070
Stynes, Martin
2003
Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem. Zbl 1200.35046
Linß, Torsten; Stynes, Martin
2001
A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem. Zbl 1012.65076
Kopteva, Natalia; Stynes, Martin
2001
Numerical methods on Shishkin meshes for linear convection-diffusion problems. Zbl 0988.76062
Linß, Torsten; Stynes, Martin
2001
The SDFEM on Shishkin meshes for linear convection-diffusion problems. Zbl 0969.65106
Linß, Torsten; Stynes, Martin
2001
Approximation of derivatives in a convection-diffusion two-point boundary value problem. Zbl 0987.65069
Kopteva, Natalia; Stynes, Martin
2001
The streamline-diffusion method for nonconforming $$Q_1^{\text{rot}}$$ elements on rectangular tensor-product meshes. Zbl 0980.65125
Stynes, Martin; Tobiska, Lutz
2001
Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers. Zbl 0986.65109
Stynes, M.; Tobiska, L.
2001
$$n$$-widths and singularly perturbed boundary value problems. II. Zbl 1097.35013
Kellogg, R. Bruce; Stynes, Martin
2001
A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems. Zbl 0939.65117
Linß, Torsten; Stynes, Martin
1999
n-widths and singularly perturbed boundary value problems. Zbl 0974.41018
Kellogg, R. Bruce; Stynes, Martin
1999
On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Zbl 0934.65127
Franca, L. P.; Nesliturk, A.; Stynes, M.
1998
...and 46 more Documents
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### Cited by 1,692 Authors

 94 Stynes, Martin 45 Natesan, Srinivasan 43 O’Riordan, Eugene 41 Roos, Hans-Görg 35 Wang, Hong 33 Gracia, José Luis 28 Zheng, Xiangcheng 27 Cen, Zhongdi 27 Shishkin, Grigoriĭ Ivanovich 26 Kadalbajoo, Mohan K. 26 Kopteva, Natalia Viktorovna 25 Linß, Torsten 24 Franz, Sebastian 23 Clavero, Carmelo 23 Liu, Xiaowei 22 Kumar, Sunil 22 Zhang, Jin 21 John, Volker 21 Patidar, Kailash C. 20 Liu, Li-Bin 18 Kaushik, Aditya 18 Zhang, Zhimin 17 Huang, Chaobao 17 Jung, Changyeol 17 Tamilselvan, Ayyadurai 16 Chen, Yanping 16 Duressa, Gemechis File 16 Xu, Aimin 15 Amiraliyev, Gabil M. 15 Kumar, Mukesh 15 Tobiska, Lutz 14 Madden, Niall 14 Rao, S. Chandra Sekhara 14 Temam, Roger Meyer 13 Chen, Hu 13 Le, Anbo 13 Liu, Fawang 13 Matthies, Gunar 13 Xenophontos, Christos A. 12 Knobloch, Petr 12 Teofanov, Ljiljana 11 Kellogg, Bruce 11 Li, Jichun 11 Liao, HongLin 11 Lu, Shujuan 11 Mohapatra, Jugal 11 Nesliturk, Ali I. 11 Nhan, Thái Anh 11 Zaky, Mahmoud A. 11 Zarin, Helena 10 Çakır, Musa 10 Das, Pratibhamoy 10 Jorge, Juan Carlos 10 Li, Dongfang 10 Miller, John James Henry 10 Munyakazi, Justin B. 10 Vulanović, Relja 9 An, Na 9 Franca, Leopoldo Luis Cabo Penna 9 Fu, Hongfei 9 Gupta, Vikas 9 Hendy, Ahmed S. 9 Meng, XiangYun 9 Mukherjee, Kaushik 9 Valanarasu, T. 9 Valarmathi, Sigamani 9 Yan, Yubin 9 Zhang, Jiwei 9 Zhou, Zhi 9 Zhu, Peng 8 Barrenechea, Gabriel R. 8 Bu, Weiping 8 Erdogan, Fevzi 8 Huang, Jian 8 Novo, Julia 8 Sharma, Kapil Kumar 8 Sharma, Manju 8 Sumit, S. 8 Uzelac, Zorica 8 Valentin, Frédéric 8 Woldaregay, Mesfin Mekuria 7 Bawa, Rajesh Kumar 7 Chen, Shaochun 7 Huang, Zhongyi 7 Jia, Jinhong 7 Jin, Bangti 7 Li, Hong 7 Linke, Alexander 7 Lisbona Cortés, Francisco Javier 7 Liu, Yang 7 Lyu, Pin 7 Sendur, Ali 7 Shi, Dongyang 7 Sun, Zhizhong 7 Vong, Seakweng 7 Wang, Song 7 Xu, Jinchao 7 Yang, Xuehua 7 Yin, Baoli 7 Zadorin, Aleksandr Ivanovich ...and 1,592 more Authors
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### Cited in 214 Serials

 131 Journal of Computational and Applied Mathematics 123 Applied Mathematics and Computation 114 Journal of Scientific Computing 103 Computers & Mathematics with Applications 95 Applied Numerical Mathematics 84 Computer Methods in Applied Mechanics and Engineering 74 Numerical Algorithms 58 International Journal of Computer Mathematics 49 Applied Mathematics Letters 46 Mathematics of Computation 35 Journal of Computational Physics 33 Mathematics and Computers in Simulation 32 Advances in Computational Mathematics 30 Computational and Applied Mathematics 27 Numerische Mathematik 26 Journal of Applied Mathematics and Computing 25 BIT 24 Numerical Methods for Partial Differential Equations 23 SIAM Journal on Numerical Analysis 21 Calcolo 20 Fractional Calculus & Applied Analysis 17 Advances in Difference Equations 16 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 16 International Journal of Applied and Computational Mathematics 15 Computational Mathematics and Mathematical Physics 15 International Journal of Computational Methods 14 SIAM Journal on Scientific Computing 13 Computing 13 Journal of Optimization Theory and Applications 13 Computational Methods in Applied Mathematics 12 Journal of Mathematical Analysis and Applications 12 Communications in Nonlinear Science and Numerical Simulation 11 East Asian Journal on Applied Mathematics 10 Numerical Functional Analysis and Optimization 10 Journal of Difference Equations and Applications 10 Differential Equations and Dynamical Systems 9 Applied Mathematical Modelling 8 Applications of Mathematics 8 Mathematical Modelling and Analysis 8 AIMS Mathematics 7 Applied Mathematics and Mechanics. (English Edition) 7 Engineering Analysis with Boundary Elements 6 Journal of Complexity 6 Abstract and Applied Analysis 6 Journal of Numerical Mathematics 6 Mediterranean Journal of Mathematics 6 Advances in Applied Mathematics and Mechanics 6 Journal of Applied Mathematics & Informatics 6 Communications on Applied Mathematics and Computation 5 Journal of Differential Equations 5 Mathematical Problems in Engineering 5 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 5 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 5 Discrete and Continuous Dynamical Systems. Series B 4 Mathematical Methods in the Applied Sciences 4 Chaos, Solitons and Fractals 4 International Journal for Numerical Methods in Engineering 4 Zeitschrift für Analysis und ihre Anwendungen 4 Acta Mathematicae Applicatae Sinica. English Series 4 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 4 Turkish Journal of Mathematics 4 ETNA. Electronic Transactions on Numerical Analysis 4 Journal of Mathematical Chemistry 4 International Journal of Nonlinear Sciences and Numerical Simulation 4 Journal of Applied Mathematics 4 S$$\vec{\text{e}}$$MA Journal 4 Computational Methods for Differential Equations 4 Electronic Research Archive 3 Applicable Analysis 3 International Journal for Numerical Methods in Fluids 3 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 3 Bulletin of Mathematical Biology 3 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 3 Mathematical and Computer Modelling 3 Japan Journal of Industrial and Applied Mathematics 3 Communications in Numerical Methods in Engineering 3 Mathematical Communications 3 Journal of Systems Science and Complexity 3 Central European Journal of Mathematics 3 Hacettepe Journal of Mathematics and Statistics 3 International Journal of Numerical Analysis and Modeling 3 Boundary Value Problems 3 Proceedings of the Steklov Institute of Mathematics 3 Journal of Applied Analysis and Computation 3 Results in Applied Mathematics 2 Computers and Fluids 2 Computer Physics Communications 2 International Journal of Heat and Mass Transfer 2 Lithuanian Mathematical Journal 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Kybernetika 2 Acta Applicandae Mathematicae 2 Journal of Computational Mathematics 2 RAIRO. Modélisation Mathématique et Analyse Numérique 2 Optimization 2 Computational Mechanics 2 Journal of Integral Equations and Applications 2 Linear Algebra and its Applications 2 SIAM Review 2 Journal of Dynamics and Differential Equations ...and 114 more Serials
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### Cited in 38 Fields

 1,582 Numerical analysis (65-XX) 802 Partial differential equations (35-XX) 360 Ordinary differential equations (34-XX) 270 Fluid mechanics (76-XX) 163 Real functions (26-XX) 56 Integral equations (45-XX) 33 Calculus of variations and optimal control; optimization (49-XX) 28 Approximations and expansions (41-XX) 28 Mechanics of deformable solids (74-XX) 17 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 15 Classical thermodynamics, heat transfer (80-XX) 15 Biology and other natural sciences (92-XX) 14 Computer science (68-XX) 12 Algebraic topology (55-XX) 12 Statistical mechanics, structure of matter (82-XX) 11 Special functions (33-XX) 11 Integral transforms, operational calculus (44-XX) 11 Operator theory (47-XX) 11 Operations research, mathematical programming (90-XX) 10 Geophysics (86-XX) 10 Systems theory; control (93-XX) 8 Probability theory and stochastic processes (60-XX) 8 Optics, electromagnetic theory (78-XX) 7 Harmonic analysis on Euclidean spaces (42-XX) 6 Geometry (51-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 4 Potential theory (31-XX) 4 Functional analysis (46-XX) 3 Difference and functional equations (39-XX) 3 Convex and discrete geometry (52-XX) 3 Statistics (62-XX) 2 Quantum theory (81-XX) 2 Information and communication theory, circuits (94-XX) 1 History and biography (01-XX) 1 Field theory and polynomials (12-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Mechanics of particles and systems (70-XX)