JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Space-time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with lowand high-order IgA spaces.

[1]  Clemens Hofreither,et al.  An algorithm for low-rank approximation of bivariate functions using splines , 2017, J. Comput. Appl. Math..

[2]  M. Gander,et al.  Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems , 2016, SIAM Journal on Scientific Computing.

[3]  Helmut Gfrerer,et al.  An SQP method for mathematical programs with complementarity constraints with strong convergence properties , 2016, Kybernetika.

[4]  Olaf Steinbach,et al.  Space-Time Finite Element Methods for Parabolic Problems , 2015, Comput. Methods Appl. Math..

[5]  Boris S. Mordukhovich,et al.  Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions , 2015, SIAM J. Optim..

[6]  Ulrich Langer,et al.  Shape Optimization of an Electric Motor Subject to Nonlinear Magnetostatics , 2015, SIAM J. Sci. Comput..

[7]  Bert Jüttler,et al.  Geometry + Simulation Modules: Implementing Isogeometric Analysis , 2014 .

[8]  Ulrich Langer,et al.  Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems , 2014, Comput. Methods Appl. Math..

[9]  Alfio Quarteroni,et al.  Isogeometric Analysis and error estimates for high order partial differential equations in Fluid Dynamics , 2014 .

[10]  Christoph Lehrenfeld,et al.  The Nitsche XFEM-DG Space-Time Method and its Implementation in Three Space Dimensions , 2014, SIAM J. Sci. Comput..

[11]  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..

[12]  Maxim A. Olshanskii,et al.  Error Analysis of a Space-Time Finite Element Method for Solving PDEs on Evolving Surfaces , 2014, SIAM J. Numer. Anal..

[13]  Karsten Urban,et al.  An improved error bound for reduced basis approximation of linear parabolic problems , 2013, Math. Comput..

[14]  Randolph E. Bank,et al.  An error analysis of some higher order space-time moving finite elements , 2013, Comput. Vis. Sci..

[15]  Maxim A. Olshanskii,et al.  An Eulerian Space-Time Finite Element Method for Diffusion Problems on Evolving Surfaces , 2013, SIAM J. Numer. Anal..

[16]  Thomas J. R. Hughes,et al.  Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements , 2013, Numerische Mathematik.

[17]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[18]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[19]  Olaf Steinbach,et al.  Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods , 2011, Comput. Vis. Sci..

[20]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[21]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[22]  Rob P. Stevenson,et al.  Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..

[23]  Marek Behr,et al.  Simplex space–time meshes in finite element simulations , 2008 .

[24]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[25]  Tayfun E. Tezduyar,et al.  Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces , 2006 .

[26]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[27]  Joachim Schöberl,et al.  Numerical analysis of nonlinear multiharmonic eddy current problems , 2005, Numerische Mathematik.

[28]  Tayfun E. Tezduyar,et al.  Enhanced-discretization space time technique (EDSTT) , 2004 .

[29]  J. V. D. Vegt,et al.  Space--time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. general formulation , 2002 .

[30]  Thomas J. R. Hughes,et al.  A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems , 1997 .

[31]  Peter Hansbo,et al.  Space-time oriented streamline diffusion methods for nonlinear conservation laws in one dimension , 1994 .

[32]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[33]  S. Mittal,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II: Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , 1992 .

[34]  T. Janik,et al.  The h‐p version of the finite element method for parabolic equations. II. The h‐p version in time , 1990 .

[35]  Sotoshi Yamada,et al.  Harmonic field calculation by the combination of finite element analysis and harmonic balance method , 1988 .

[36]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[37]  Alexander Ostermann,et al.  Multi-grid dynamic iteration for parabolic equations , 1987 .

[38]  Jukka Saranen,et al.  Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations , 1986 .

[39]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[40]  W. Mayer,et al.  Partielle Differentialgleichungen , 1930 .

[41]  Bernd Eggers,et al.  Nonlinear Functional Analysis And Its Applications , 2016 .

[42]  Martin J. Gander,et al.  50 Years of Time Parallel Time Integration , 2015 .

[43]  U. Langer,et al.  Analysis of Discontinuous Galerkin IGA Approximations to Elliptic Boundary Value Problems , 2014 .

[44]  J. Schöberl,et al.  Analysis of a Time Multigrid Algorithm for DG-Discretizations in Time , 2014 .

[45]  Roman Andreev,et al.  Stability of sparse space–time finite element discretizations of linear parabolic evolution equations , 2013 .

[46]  Olaf Steinbach,et al.  A DG Space-Time Domain Decomposition Method , 2013, Domain Decomposition Methods in Science and Engineering XX.

[47]  Martin Neumüller,et al.  Space-Time Methods: Fast Solvers and Applications , 2013 .

[48]  Rob P. Stevenson,et al.  Adaptive Wavelet Schemes for Parabolic Problems: Sparse Matrices and Numerical Results , 2011, SIAM J. Numer. Anal..

[49]  Fu Xiaojin,et al.  Isogeometric Analysis Toward Integration of CAD and CAE , 2011 .

[50]  Graham Horton,et al.  Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods , 2005, Computing.

[51]  Jens Lang,et al.  Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems - Theory, Algorithm, and Applications , 2001, Lecture Notes in Computational Science and Engineering.

[52]  G. Gustafson,et al.  Boundary Value Problems of Mathematical Physics , 1998 .

[53]  W. Zulehner,et al.  A Finite Element Solver for a Multiharmonic Parabolic Optimal Control Problem , 1997 .

[54]  Craig C. Douglas,et al.  A Rigorous Analysis of Time Domain Parallelism , 1995, Parallel Algorithms Appl..

[55]  S. Vandewalle.  Parallel multigrid waveform relaxation for parabolic problems , 1993 .

[56]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[57]  Ulrich Langer,et al.  JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics A Robust Preconditioned-MinRes-Solver for Distributed Time-Periodic Eddy Current Optimal Control , 2011 .