×

Unified form language: a domain-specific language for weak formulations of partial differential equations. (English) Zbl 1308.65175


MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D25 Numerical differentiation
68W30 Symbolic computation and algebraic computation
68N15 Theory of programming languages
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
65Y04 Numerical algorithms for computer arithmetic, etc.
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] C. Abert, L. Exl, G. Selke, A. Drews, and T. Schrefl. 2012. Numerical methods for the strayfield calculation: A comparison of recently developed algorithms. J. Magn. Magn. Mater.
[2] M. S. Alnæs. 2009. A compiler framework for automatic linearization and efficient discretization of nonlinear partial differential equations. PhD thesis, University of Oslo.
[3] M. S. Alnæs and K.-A. Mardal. 2010. On the efficiency of symbolic computations combined with code generation for finite element methods. ACM Trans. Math. Softw. 37, 1. · Zbl 1364.68375
[4] M. S. Alnæs and K.-A. Mardal. 2012. SyFi and SFC: Symbolic finite elements and form compilation. In Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, and G. N. Wells, Eds., Lecture Notes in Computational Science and Engineering, vol. 84, chapter 15, Springer.
[5] M. S. Alnæs, A. Logg, K.-A. Mardal, O. Skavhaug, and H. P. Langtangen. 2009. Unified framework for finite element assembly. Int. J. Comput. Sci. Eng. 4, 4, 231–244. http://dx.doi.org/10.1504/IJCSE.2009.029160.
[6] M. S. Alnæs, A. Logg, and K.-A. Mardal. 2012. UFC: a finite element code generation interface. In Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, and G. N. Wells, Eds., Lecture Notes in Computational Science and Engineering, vol. 84, chapter 16, Springer.
[7] D. N. Arnold. 1982. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 4, 742–760. · Zbl 0482.65060
[8] D. N. Arnold, F. Brezzi, and M. Fortin. A stable finite element for the Stokes equations. Calcolo 21, 4, 337–344. http://dx.doi.org/10.1007/BF02576171. · Zbl 0593.76039
[9] D. N. Arnold, R. S. Falk, and R. Winther. 2006. Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155. · Zbl 1185.65204
[10] D. N. Arnold, R. S. Falk, J. Guzmán, and G. Tsogtgerel. 2012. On the consistency of the combinatorial codifferential. arXiv preprint arXiv:1212.4472. · Zbl 1298.58003
[11] B. Bagheri and L. R. Scott. About Analysa. 2004. Tech. Rep. TR-2004-09, Department of Computer Science, University of Chicago. http://www.cs.uchicago.edu/research/publications/techreports/TR-2004-09.
[12] G. Baumgartner, A. Auer, et al. 2005. Synthesis of high-performance parallel programs for a class of ab Initio quantum chemistry models. Proc. IEEE 93, 2, 276–292. http://dx.doi.org/10.1109/JPROC.2004.840311.
[13] Boost. 2012. Boost C++ libraries, http://www.boost.org/.
[14] C. Brandenburg, F. Lindemann, M. Ulbrich, and S. Ulbrich. 2012. Advanced numerical methods for pde constrained optimization with application to optimal design in Navier Stokes flow. In Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, 257–275. · Zbl 1356.49019
[15] F. Brezzi, J. Douglas, and L. D. Marini. 1985. Two families of mixed elements for second order elliptic problems. Numer. Math. 47, 217–235. · Zbl 0599.65072
[16] P. Brunner, C. Clason, M. Freiberger, and H. Scharfetter. 2012. A deterministic approach to the adapted optode placement for illumination of highly scattering tissue. Biomed. Optics Express 3, 7, 1732.
[17] DOLFIN. 2012. DOLFIN library. http://launchpad.net/dolfin.
[18] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes. 2012. Automated derivation of the adjoint of high-level transient finite element programs. http://arxiv.org/abs/1204.5577. · Zbl 1362.65103
[19] S. W. Funke and P. E. Farrell. 2013. A framework for automated pde-constrained optimisation. CoRR, abs/1302.3894.
[20] E. Gamma, R. Helm, R. Johnson, and J. Vlissides. 1993. Design patterns: Abstraction and reuse of object-oriented design. In Proceedings of the European Conference on Object-Oriented Programming. 406–431.
[21] A. Griewank. 1989. On automatic differentiation. In Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, Eds., Kluwer, 83–108. · Zbl 0696.65015
[22] J. Hake, A. G. Edwards, Z. Yu, P. M. Kekenes-Huskey, A. P. Michailova, J. A. McCammon, M. J. Holst, M. Hoshijima, and A. D. McCulloch. 2012. Modelling cardiac calcium sparks in a three-dimensional reconstruction of a calcium release unit. J. Physiol. 590, 18, 4403–4422.
[23] P. Hudak. 1996. Building domain-specific embedded languages. ACM Comput. Surv. 28, 196.
[24] J. Karczmarczuk. 2001. Functional differentiation of computer programs. Higher-Order and Symbolic Computation, 14, 1, 35–57. http://dx.doi.org/10.1023/A:1011501232197. · Zbl 0967.68174
[25] R. C. Kirby and A. Logg. 2006. A compiler for variational forms. ACM Trans. Math. Softw. 32, 3. http://dx.doi.org/10.1145/1163641.1163644. · Zbl 05458453
[26] R. C. Kirby and A. Logg. 2012. Finite element variational forms. In Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, and G. N. Wells, Eds., Lecture Notes in Computational Science and Engineering, vol. 84, chapter 5, Springer.
[27] J. Korelc. 1997. Automatic generation of finite-element code by simultaneous optimization of expressions. Theor. Comput. Sci. 187, 1–2, 231–248. http://dx.doi.org/10.1016/S0304-3975(97)00067-4. · Zbl 0893.68084
[28] R. J. Labeur and G. N. Wells. 2012. Energy stable and momentum conserving hybrid finite element method for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 34, 2, A889–A913. http://dx.doi.org/10.1137/100818583. · Zbl 1391.76344
[29] A. Logg and G. N. Wells. 2010. DOLFIN: Automated finite element computing. ACM Trans. Math. Softw. 37, 2, 20:1–20:28. http://dx.doi.org/10.1145/1731022.1731030. · Zbl 1364.65254
[30] A. Logg, K.-A. Mardal, and G. N. Wells, Eds. 2012a. Automated Solution of Differential Equations by the Finite Element Method. Lecture Notes in Computational Science and Engineering, vol. 84, Springer. · Zbl 1247.65105
[31] A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells. 2012b. FFC: The FEniCS form compiler. In Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, and G. N. Wells, Eds., Lecture Notes in Computational Science and Engineering, vol. 84, chapter 11, Springer.
[32] A. Logg, G. N. Wells, and J. Hake. 2012c/ DOLFIN: A C++/Python finite element library. In Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, and G. N. Wells, Eds., Lecture Notes in Computational Science and Engineering, vol. 84, Chapter 10, Springer.
[33] K. Long, R. Kirby, and B. Van Bloemen Waanders. 2010. Unified embedded parallel finite element computations via software-based Fréchet differentiation. SIAM J. Sci. Comput. 32, 3323–3351. · Zbl 1221.65306
[34] M. Maraldi, G. N. Wells, and L. Molari. 2011. Phase field model for coupled displacive and diffusive microstructural processes under thermal loading. J. Mech. Phys. Solids 59, 8, 1596–1612. http://dx.doi.org/10.1016/j.jmps.2011.04.017. · Zbl 1270.74159
[35] G. Markall, A. Slemmer, D. Ham, P. Kelly, C. Cantwell, and S. Sherwin. 2012. Finite element assembly strategies on multi-core and many-core architectures. Int. J. Numer. Methods Fluids. http://dx.doi.org/10.1002/fld.3648.
[36] G. R. Markall, D. A. Ham, and P. H. J. Kelly. 2010. Towards generating optimised finite element solvers for GPUs from high-level specifications. Proc. Comp. Sci. 1, 1, 1815–1823.
[37] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. 2012. A stabilized Nitsche fictitious domain method for the Stokes problem. http://arxiv.org/abs/1206. 1933. · Zbl 1417.76028
[38] M. Mortensen, H. P. Langtangen, and G. N. Wells. 2011. A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations. Adv. Water Res. 34, 9, 1082–1101, 2011. http://dx.doi.org/j.advwatres.2011.02.013.
[39] J. C. Nédélec. 1986. A new family of mixed finite elements in R3. Numer. Math. 50, 1, 57–81.
[40] M. Nikbakht and G. N. Wells. 2009. Automated modelling of evolving discontinuities. 2, 3, 1008–1030. http://dx.doi.org/10.3390/a2031008. · Zbl 1461.65252
[41] K. B. Ølgaard and G. N. Wells. 2010. Optimisations for quadrature representations of finite element tensors through automated code generation. ACM Trans. Math. Softw. 37, 1, 8:1–8:23.
[42] K. B. Ølgaard, A. Logg, and G. N. Wells. 2008. Automated code generation for discontinuous Galerkin methods. SIAM J. Sci. Comput. 31, 2, 849–864. http://dx.doi.org/10.1137/070710032. · Zbl 1189.65283
[43] B. A. Pearlmutter and J. M. Siskind. 2007. Lazy multivariate higher-order forward-mode AD. In Proceedings of the 34th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. 155–160. http://dx.doi.org/10.1145/1190216.1190242. · Zbl 1295.65028
[44] C. Prud’homme. 2006. A domain specific embedded language in C++ for automatic differentiation, projection, integration and variational formulations. In Scientific Programming 14, 81–110.
[45] C. Prud’homme. 2011. Feel++. https://forge.imag.fr/projects/life/.
[46] PyChecker. 2011. PyChecker: A Python source code checking tool. http://pychecker.sourceforge.net.
[47] M. E. Rognes and A. Logg. 2012. Automated goal-oriented error control I: stationary variational problems. http://arxiv.org/abs/1204.6643. · Zbl 1276.65071
[48] M. E. Rognes, R. C. Kirby, and A. Logg. 2009. Efficient assembly of H(div) and H(curl) conforming finite elements. SIAM J. Sci. Comput. 31, 6, 4130–4151. http://dx.doi.org/10.1137/08073901X. · Zbl 1206.65248
[49] E. Rosseel and G. N. Wells. 2012. Optimal control with stochastic PDE constraints and uncertain controls. Comput. Meth. Appl. Mech. Eng. 213–216, 152–167. http://dx.doi.org/10.1016/j.cma.2011.11.026. · Zbl 1243.49034
[50] J. R. Shewchuk and O. Ghattas. 1993. A compiler for parallel finite element methods with domain-decomposed unstructured meshes. In Proceedings of the 7th International Conference on Domain Decomposition Methods in Scientific and Engineering Computing, D. E. Keyes and J. Xu, Eds., Contemporary Mathematics, volume 180, American Mathematical Society, 445–450. · Zbl 0817.65079
[51] J. M. Siskind and B. A. Pearlmutter. 2008. Nesting forward-mode AD in a functional framework. Higher Order Symbol. Comput. 21, 4, 361–376. http://dx.doi.org/10.1007/s10990-008-9037-1. · Zbl 1175.68104
[52] C. Taylor and P. Hood. 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids. 1, 1, 73–100. · Zbl 0328.76020
[53] UFL. Unified Form Language, 2012. http://launchpad.net/ufl.
[54] P. Wang. 1986. FINGER: A symbolic system for automatic generation of numerical programs in finite element analysis. J. Symbolic Comput. 2, 3, 305–316. http://dx.doi.org/10.1016/S0747-7171(86)80029-3. · Zbl 0604.65078
[55] G. N. Wells. 2011. Analysis of an interface stabilised finite element method: The advection-diffusion-reaction equation. SIAM J. Numer. Anal. 49, 1, 87–109. http://dx.doi.org/10.1137/090775464. · Zbl 1226.65097
[56] J. Xiong, J. Johnson, R. Johnson, and D. Padua. 2001. SPL: A language and compiler for DSP algorithms. SIGPLAN Not. 36, 298–308. http://dx.doi.org/10.1145/381694.378860.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.