7. Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution
暂无分享,去创建一个
[1] Dmitriy Leykekhman,et al. A posteriori error estimates by recovered gradients in parabolic finite element equations , 2008 .
[2] Martin Vohralík,et al. A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations , 2007, SIAM J. Numer. Anal..
[3] H. Freudenthal. Simplizialzerlegungen von Beschrankter Flachheit , 1942 .
[4] O. Lakkis,et al. Gradient recovery in adaptive finite-element methods for parabolic problems , 2009, 0905.2764.
[5] Igor Kossaczký. A recursive approach to local mesh refinement in two and three dimensions , 1994 .
[6] G. Murali Mohan Reddy,et al. A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems , 2016, J. Sci. Comput..
[7] Omar Lakkis,et al. A comparison of duality and energy a posteriori estimates for L∞(0, T;L2(Ω)) in parabolic problems , 2007, Math. Comput..
[8] Lothar Banz,et al. hp-adaptive IPDG/TDG-FEM for parabolic obstacle problems , 2014, Comput. Math. Appl..
[9] M. Chipot. Finite Element Methods for Elliptic Problems , 2000 .
[10] Claes Johnson,et al. Error estimates and automatic time step control for nonlinear parabolic problems, I , 1987 .
[11] Bärbel Holm,et al. Fully reliable error control for first-order evolutionary problems , 2017, Comput. Math. Appl..
[12] F. Tröltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .
[13] Boris Vexler,et al. Pointwise Best Approximation Results for Galerkin Finite Element Solutions of Parabolic Problems , 2015, SIAM J. Numer. Anal..
[14] Olaf Steinbach,et al. Coercive space-time finite element methods for initial boundary value problems , 2020 .
[15] Ricardo H. Nochetto,et al. A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality , 2004 .
[16] K. Stüben. A review of algebraic multigrid , 2001 .
[17] Ludmil T. Zikatanov,et al. A monotone finite element scheme for convection-diffusion equations , 1999, Math. Comput..
[18] R. Verfürth. A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations , 1994 .
[19] J. Oden,et al. A Posteriori Error Estimation in Finite Element Analysis , 2000 .
[20] U. Trottenberg,et al. A note on MGR methods , 1983 .
[21] Dietrich Braess,et al. Equilibrated residual error estimates are p-robust , 2009 .
[22] Jesse Chan,et al. Locally conservative discontinuous Petrov-Galerkin finite elements for fluid problems , 2014, Comput. Math. Appl..
[23] Ricardo H. Nochetto,et al. A posteriori error estimation and adaptivity for degenerate parabolic problems , 2000, Math. Comput..
[24] A. Schmidt,et al. Design and convergence analysis for an adaptive discretization of the heat equation , 2012 .
[25] D. Aronson,et al. Non-negative solutions of linear parabolic equations , 1968 .
[26] Santiago Badia,et al. Space-Time Balancing Domain Decomposition , 2017, SIAM J. Sci. Comput..
[27] J. Lions,et al. Sur les problèmes mixtes pour certains systèmes paraboliques dans les ouverts non cylindriques , 1957 .
[28] R. Hoppe,et al. A Posteriori Estimates for Cost Functionals of Optimal Control Problems , 2006 .
[29] A. Friedman. Variational principles and free-boundary problems , 1982 .
[30] Eberhard Bänsch,et al. Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..
[31] Rolf Rannacher,et al. Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..
[32] Denis Devaud. hp-approximation of linear parabolic evolution problems in H^{1/2} , 2017 .
[33] Philippe Destuynder,et al. Explicit error bounds in a conforming finite element method , 1999, Math. Comput..
[34] Ricardo H. Nochetto,et al. A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis , 2013, Found. Comput. Math..
[35] Christian Mollet,et al. Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems , 2014, Comput. Methods Appl. Math..
[36] Rolf Krause,et al. Variational space–time elements for large-scale systems , 2017 .
[37] L. R. Scott,et al. Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .
[38] Douglas N. Arnold,et al. Finite element exterior calculus for parabolic problems , 2012, 1209.1142.
[39] Joseph M. Maubach,et al. Local bisection refinement for $n$-simplicial grids generated by reflection , 2017 .
[40] Simona Perotto,et al. New anisotropic a priori error estimates , 2001, Numerische Mathematik.
[41] T. Hughes,et al. Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .
[42] Alan Demlow,et al. Sharply local pointwise a posteriori error estimates for parabolic problems , 2010, Math. Comput..
[43] Martin Neumüller,et al. Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems , 2018 .
[44] TZ , 2019, Springer Reference Medizin.
[45] Jörg Grande,et al. Red-green refinement of simplicial meshes in d dimensions , 2018, Math. Comput..
[46] Kenneth Eriksson,et al. Time discretization of parabolic problems by the discontinuous Galerkin method , 1985 .
[47] V. Bulgakov. Multi-level iterative technique and aggregation concept with semi-analytical preconditioning for solving boundary-value problems , 1993 .
[48] Norbert Heuer,et al. A Time-Stepping DPG Scheme for the Heat Equation , 2016, Comput. Methods Appl. Math..
[49] Jay Gopalakrishnan. A CLASS OF DISCONTINUOUS PETROV-GALERKIN METHODS. PART II: OPTIMAL TEST FUNCTIONS , 2009 .
[50] R. Nochetto,et al. Theory of adaptive finite element methods: An introduction , 2009 .
[51] Andreas Veeser,et al. A posteriori error estimators, gradient recovery by averaging, and superconvergence , 2006, Numerische Mathematik.
[52] Olaf Steinbach,et al. Space-Time Finite Element Methods for Parabolic Problems , 2015, Comput. Methods Appl. Math..
[53] Bo Li,et al. Analysis of a Class of Superconvergence Patch Recovery Techniques for Linear and Bilinear Finite Elements , 1999 .
[54] Peter Monk,et al. Continuous finite elements in space and time for the heat equation , 1989 .
[55] Leszek Demkowicz,et al. A class of discontinuous Petrov-Galerkin methods. Part III , 2012 .
[56] Graham Horton,et al. A Space-Time Multigrid Method for Parabolic Partial Differential Equations , 1995, SIAM J. Sci. Comput..
[58] F. Bornemann,et al. Adaptive multivlevel methods in three space dimensions , 1993 .
[59] Fredi Tröltzsch,et al. On the optimal control of the Schlögl-model , 2013, Comput. Optim. Appl..
[60] Panayot S. Vassilevski,et al. On Generalizing the Algebraic Multigrid Framework , 2004, SIAM J. Numer. Anal..
[61] ROB STEVENSON,et al. The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..
[62] Leszek F. Demkowicz,et al. Analysis of the DPG Method for the Poisson Equation , 2011, SIAM J. Numer. Anal..
[63] Boris Vexler,et al. Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems , 2007, SIAM J. Control. Optim..
[64] Svetlana Matculevich,et al. Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation , 2013, Appl. Math. Comput..
[65] Barry Joe,et al. Quality Local Refinement of Tetrahedral Meshes Based on Bisection , 1995, SIAM J. Sci. Comput..
[66] Charalambos Makridakis,et al. Space and time reconstructions in a posteriori analysis of evolution problems , 2007 .
[67] O. Steinbach,et al. Space–time DG methods for the coupled electro–mechanical activation of the human heart , 2014 .
[68] Serge Prudhomme,et al. A posteriori error estimation and error control for finite element approximations of the time-dependent Navier-Stokes equations , 1999 .
[69] Andrew Gillette,et al. Finite Element Exterior Calculus for Evolution Problems , 2012, 1202.1573.
[70] M. Picasso. Adaptive finite elements for a linear parabolic problem , 1998 .
[71] Robert D. Falgout,et al. Compatible Relaxation and Coarsening in Algebraic Multigrid , 2009, SIAM J. Sci. Comput..
[72] M. Holst,et al. Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces , 2015, Journal of Computational Mathematics.
[73] Stefano Berrone,et al. A Local-in-Space-Timestep Approach to a Finite Element Discretization of the Heat Equation with a Posteriori Estimates , 2009, SIAM J. Numer. Anal..
[74] Arnold Reusken,et al. Parallel Multilevel Tetrahedral Grid Refinement , 2005, SIAM J. Sci. Comput..
[75] Martin Vohralík,et al. A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation , 2010, SIAM J. Numer. Anal..
[76] Christoph Schwab,et al. Space–time hp-approximation of parabolic equations , 2018, Calcolo.
[77] F. Schlögl,et al. A characteristic critical quantity in nonequilibrium phase transitions , 1983 .
[78] C. Kreuzer. Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic $$p$$-Laplacian , 2013 .
[79] Javier de Frutos,et al. A posteriori error estimation with the p-version of the finite element method for nonlinear parabolic differential equations☆ , 2002 .
[80] C. T. Traxler,et al. An algorithm for adaptive mesh refinement inn dimensions , 1997, Computing.
[81] William L. Briggs,et al. A multigrid tutorial , 1987 .
[82] Ningning Yan,et al. A posteriori error estimates for optimal control problems governed by parabolic equations , 2003, Numerische Mathematik.
[83] Magnus Fontes,et al. Parabolic equations with low regularity , 1996 .
[84] Alan Demlow,et al. A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems , 2007, SIAM J. Numer. Anal..
[85] Shinichi Kawahara. Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain , 1977 .
[86] Alexandre Ern,et al. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations , 2007, Math. Comput..
[87] Sergey I. Repin,et al. A posteriori error estimation for variational problems with uniformly convex functionals , 2000, Math. Comput..
[88] Martin J. Gander,et al. 50 Years of Time Parallel Time Integration , 2015 .
[89] G. Gustafson,et al. Boundary Value Problems of Mathematical Physics , 1998 .
[90] Ulrich Langer,et al. Multipatch Space-Time Isogeometric Analysis of Parabolic Diffusion Problems , 2017, LSSC.
[91] Stig Larsson,et al. Compressive space-time Galerkin discretizations of parabolic partial differential equations , 2015, 1501.04514.
[92] Wolfgang Dahmen,et al. Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.
[93] Marian Brezina,et al. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems , 2005, Computing.
[94] Miss A.O. Penney. (b) , 1974, The New Yale Book of Quotations.
[95] U. Langer,et al. A posteriori error estimates for space-time IgA approximations to parabolic initial boundary value problems , 2016, 1612.08998.
[96] Ricardo H. Nochetto,et al. Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems , 2006, Math. Comput..
[97] Xianjuan Li,et al. Finite difference/spectral approximations for the fractional cable equation , 2010, Math. Comput..
[98] P. Wesseling. An Introduction to Multigrid Methods , 1992 .
[99] Dietrich Braess,et al. Equilibrated residual error estimator for edge elements , 2007, Math. Comput..
[100] J. W. Ruge,et al. 4. Algebraic Multigrid , 1987 .
[101] H. Rentz-Reichert,et al. UG – A flexible software toolbox for solving partial differential equations , 1997 .
[102] Constantin Popa. ALGEBRAIC MULTIGRID SMOOTHING PROPERTY OF KACZMARZ'S RELAXATION FOR GENERAL RECTANGULAR LINEAR SYSTEMS , 2007 .
[103] Kenneth Eriksson,et al. Adaptive Finite Element Methods for Parabolic Problems VI: Analytic Semigroups , 1998 .
[104] T. Hughes,et al. Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .
[105] Wenbin Liu,et al. A Posteriori Error Estimates for Finite Element Approximation of Parabolic p-Laplacian , 2006, SIAM J. Numer. Anal..
[106] R. Rodríguez. Some remarks on Zienkiewicz‐Zhu estimator , 1994 .
[107] Huidong Yang,et al. Partitioned solvers for the fluid-structure interaction problems with a nearly incompressible elasticity model , 2011, Comput. Vis. Sci..
[108] Douglas N. Arnold,et al. Locally Adapted Tetrahedral Meshes Using Bisection , 2000, SIAM Journal on Scientific Computing.
[109] Rolf Rannacher,et al. Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .
[110] D. Arnold. Finite Element Exterior Calculus , 2018 .
[111] Yves Achdou,et al. A Posteriori Error Estimates for Parabolic Variational Inequalities , 2008, J. Sci. Comput..
[112] Martin J. Gander,et al. Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems , 2014, SIAM J. Sci. Comput..
[113] A posteriori error estimates for approximations of evolutionary convection–diffusion problems , 2010, 1012.5089.
[114] Endre Süli,et al. Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.
[115] Randolph E. Bank,et al. PLTMG - a software package for solving elliptic partial differential equations: users' guide 8.0 , 1998, Software, environments, tools.
[116] Xuemin Tu. Three-Level BDDC in Three Dimensions , 2007, SIAM J. Sci. Comput..
[117] Panayot S. Vassilevski,et al. The Construction of the Coarse de Rham Complexes with Improved Approximation Properties , 2014, Comput. Methods Appl. Math..
[118] Sören Bartels,et al. Quasi-optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities , 2011, Math. Comput..
[119] Ulrich Langer,et al. Space–time isogeometric analysis of parabolic evolution problems , 2015, 1509.02008.
[120] J. Lions,et al. Non-homogeneous boundary value problems and applications , 1972 .
[121] Stefano Berrone,et al. Adaptive discretization of stationary and incompressible Navier–Stokes equations by stabilized finite element methods , 2001 .
[122] Carsten Carstensen,et al. Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..
[123] Sergey Repin,et al. Accuracy Verification Methods , 2014 .
[124] Ludmil T. Zikatanov,et al. Arbitrary dimension convection-diffusion schemes for space-time discretizations , 2016, J. Comput. Appl. Math..
[125] R. Bank,et al. Some Refinement Algorithms And Data Structures For Regular Local Mesh Refinement , 1983 .
[126] Robert D. Falgout,et al. Multigrid methods with space–time concurrency , 2017, Comput. Vis. Sci..
[127] Mats Boman,et al. Global and Localised A Posteriori Error Analysis in the maximum norm for finite element approximations of a convection-diffusion problem , 2000 .
[128] Haijun Wu,et al. A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow , 2005, J. Sci. Comput..
[129] Karsten Urban,et al. An improved error bound for reduced basis approximation of linear parabolic problems , 2013, Math. Comput..
[130] Theodoros Katsaounis,et al. A posteriori error control and adaptivity for Crank–Nicolson finite element approximations for the linear Schrödinger equation , 2013, Numerische Mathematik.
[131] L. Caffarelli,et al. An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.
[132] Ferdinand Kickinger,et al. Algebraic Multi-grid for Discrete Elliptic Second-Order Problems , 1998 .
[133] J. Douglas,et al. Galerkin Methods for Parabolic Equations , 1970 .
[134] Panagiotis Chatzipantelidis,et al. A Posteriori Error Estimates for the Two-Step Backward Differentiation Formula Method for Parabolic Equations , 2010, SIAM J. Numer. Anal..
[135] Omar Lakkis,et al. Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems , 2006, Math. Comput..
[136] Jia Feng,et al. An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems , 2004, Math. Comput..
[137] Fredi Tröltzsch,et al. Sparse Optimal Control of the Schlögl and FitzHugh–Nagumo Systems , 2013, Comput. Methods Appl. Math..
[138] Thomas A. Manteuffel,et al. Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study , 2017, SIAM J. Sci. Comput..
[139] Martin Vohralík,et al. Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems , 2010, J. Comput. Appl. Math..
[140] Rob P. Stevenson,et al. Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..
[141] Ricardo H. Nochetto,et al. Error estimates for multidimensional singular parabolic problems , 1987 .
[142] D. Brandt,et al. Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .
[143] Bosco Garc,et al. Postprocessing the Galerkin Method: The Finite-Element Case , 1999 .
[144] Roland Becker,et al. A posteriori error estimation for finite element discretization of parameter identification problems , 2004, Numerische Mathematik.
[145] Roman Andreev,et al. Stability of space-time Petrov-Galerkin discretizations for parabolic evolution equations , 2012 .
[146] Rolf Rannacher,et al. An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.
[147] J. Bey,et al. Tetrahedral grid refinement , 1995, Computing.
[148] Carsten Carstensen,et al. Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..
[149] Jürgen Bey,et al. Simplicial grid refinement: on Freudenthal's algorithm and the optimal number of congruence classes , 2000, Numerische Mathematik.
[150] Yousef Saad,et al. Iterative methods for sparse linear systems , 2003 .
[151] K. Stüben,et al. Multigrid methods: Fundamental algorithms, model problem analysis and applications , 1982 .
[152] Ricardo H. Nochetto,et al. Error Control and Andaptivity for a Phase Relaxation Model , 2000 .
[153] Jinchao Xu,et al. Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..
[154] Alexandre Ern,et al. Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems , 2017, IMA Journal of Numerical Analysis.
[155] Ulrich Langer,et al. Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems , 2014, Comput. Methods Appl. Math..
[156] Francisco José Gaspar,et al. Multigrid method based on a space-time approach with standard coarsening for parabolic problems , 2018, Appl. Math. Comput..
[157] Danping Yang,et al. Sharp A Posteriori Error Estimates for Optimal Control Governed by Parabolic Integro-Differential Equations , 2014, Journal of Scientific Computing.
[158] Rüdiger Verfürth,et al. A posteriori error analysis of the fully discretized time-dependent Stokes equations , 2004 .
[159] S. Nicaise,et al. An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems , 2007 .
[160] Abner J. Salgado,et al. A Space-Time Fractional Optimal Control Problem: Analysis and Discretization , 2016, SIAM J. Control. Optim..
[161] O. Steinbach. Adaptive nite element – boundary element solution of boundary value problems , 1999 .
[162] Mark Ainsworth,et al. A Synthesis of A Posteriori Error Estimation Techniques for Conforming , Non-Conforming and Discontinuous Galerkin Finite Element Methods , 2005 .
[163] Svetlana Matculevich,et al. On the a posteriori error analysis for linear Fokker-Planck models in convection-dominated diffusion problems , 2018, Appl. Math. Comput..
[164] Simona Perotto,et al. Anisotropic error estimates for elliptic problems , 2003, Numerische Mathematik.
[165] T. Hughes,et al. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .
[166] Ohannes A. Karakashian,et al. A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..
[167] Truman Everett Ellis. Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics , 2016 .
[168] Olaf Steinbach,et al. Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods , 2011, Comput. Vis. Sci..
[169] Michael J. Holst,et al. Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces , 2010, Foundations of Computational Mathematics.
[170] Wolfgang Joppich,et al. Practical Fourier Analysis for Multigrid Methods , 2004 .
[171] Stefano Berrone,et al. Space-Time adaptive simulations for unsteady Navier-Stokes problems , 2009 .
[172] Jens Lang,et al. Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems - Theory, Algorithm, and Applications , 2001, Lecture Notes in Computational Science and Engineering.
[173] Leszek Demkowicz,et al. An Overview of the Discontinuous Petrov Galerkin Method , 2014 .
[174] Boris Vexler,et al. Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations , 2007, SIAM J. Sci. Comput..
[175] Mary Fanett. A PRIORI L2 ERROR ESTIMATES FOR GALERKIN APPROXIMATIONS TO PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS , 1973 .
[176] Yousef Saad,et al. A Greedy Strategy for Coarse-Grid Selection , 2007, SIAM J. Sci. Comput..
[177] Christine Bernardi,et al. A posteriori analysis of the finite element discretization of some parabolic equations , 2004, Math. Comput..
[178] Kenneth Eriksson,et al. Adaptive finite element methods for parabolic problems V: long-time integration , 1995 .
[179] P. Clément. Approximation by finite element functions using local regularization , 1975 .
[180] Natalia Kopteva,et al. Maximum Norm A Posteriori Error Estimation for Parabolic Problems Using Elliptic Reconstructions , 2013, SIAM J. Numer. Anal..