@inbook {Simula.simula.1104,
	title = {Cerebrospinal Fluid Flow},
	booktitle = {Automated Solution of Differential Equations by the Finite Element Method},
	series = {Lecture Notes in Computational Science and Engineering},
	volume = {84},
	year = {2012},
	pages = {455-470},
	publisher = {Springer},
	organization = {Springer},
	chapter = {24},
	isbn = {978-3-642-23098-1},
	author = {St{\o}le-Hentschel, Susanne and Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre},
	editor = {Logg, Anders and Mardal, Kent-Andre and Garth N. {Wells}}
}
@article {Simula.sc.1026,
	title = {Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains},
	journal = {Journal of Scientific Computing},
	volume = {50},
	number = {1},
	year = {2012},
	pages = {198-212},
	abstract = {In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element PN {-} PN {-}2 setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P1 {-} P1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipes.},
	doi = {10.1007/s10915-011-9478-2},
	author = {Deparis, Simone and Alf Emil {L{\o}vgren}}
}
@inproceedings {Simula.simula.682,
	title = {Cyclic Cerebrospinal Fluid Flow (I): Towards the Understanding of Cerebrospinal Fluid Flow Dynamics in Chiari I Malformation},
	journal = {ASNR 49th Annual Meeting \& the Foundation of the ASNR Symposium, June 4 - 9, 2011.},
	year = {2011},
	type = {Conference},
	abstract = {Purpose Chiari I malformation leads to abnormal cerebrospinal fluid (CSF) flow dynamics and often is accompanied by syrinx formation. Phase constrast MR investigations reveal that peak flow velocities increase, whereas the altered pressure gradients are much more difficult to assess noninvasively. In this study, we use computer simulations to investigate how tonsil herniation changes pressure dynamics and flow in the cervical subarachnoid space. Materials and Methods. A 3D mathematical model of the cervical subarachnoid space, with and without herniated tonsils, was used to simulate CSF flow dynamics over several cycles of sinusoidal flow. Simulation results for the case with herniated tonsils were compared with corresponing results for normally positioned tonsils, while keeping all other parameters the same. Results Tonsil herniation caused pressure gradients to increse both in the superior-inferior direction and in axial planes of the model. The change in pressure dynamics caused a more complex flow velocity pattern with higher peak velocities, pronounced synchronous bi-directional flow, and larger velocity components within axial planes. Conclusion Chiari I malformation increases pressure gradients and flow complexity in the cervical subarachnoid space. This implies, e.g., that the spinal cord-CSF interaction dynamics changes following tonsil herniation, something which might have a role to play in syrinx formation.},
	isbn = {XXXX},
	author = {Linge, Svein and Haughton, Victor and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Helgeland, Anders and Hans Petter {Langtangen} and St{\o}verud, Karen-Helene}
}
@article {Simula.simula.24,
	title = {Effect of Tonsilar Herniation on Cyclic CSF Flow Studied With Computational Flow Analysis},
	journal = {American Journal of Neuroradiology},
	volume = {32},
	number = {5},
	year = {2011},
	pages = {1474-1481},
	abstract = {Purpose: The Chiari I malformation, characterized by tonsils extending below the foramen magnum, has increased cerebrospinal fluid (CSF) velocities compared to normal subjects. Measuring the effect of tonsilar herniation on CSF flow in humans is confounded by inter-individual variation. The goal of this study was to determine the effect of herniated tonsils on flow velocity and pressure dynamics. Materials and Methods: A previously described 3D mathematical model of the normal subarachnoid space was modified by extending the tonsils inferiorly. The chamber created was compared with the anatomy of the subarachnoid space. Pressures and velocities were calculated by Computational Fluid Analysis (CFA) methods for sinusoidal flow of a Newtonian fluid. Results were displayed as 2D color-coded plots and 3D animations. Pressure gradients and flow velocities were compared with those in the normal model. Velocity distributions were also compared with those in clinical images of CSF flow. Results: The model represented grossly the subarachnoid space of a Chiari patient. Fluid flow patterns in the Chiari model were complex with jets in some locations and stagnant flow in others. Flow jets, synchronous bi-directional flow, and pressure gradients were greater in the Chiari model than in the normal model. The distribution of flow velocities in the model corresponded well with those observed in clinical images of CSF flow in Chiari I patients. Conclusions: Tonsilar herniation per se increases the pressure gradients and the complexity of flow patterns associated with oscillatory CSF flow.},
	author = {Linge, Svein and Haughton, Victor and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Helgeland, Anders and Hans Petter {Langtangen}}
}
@article {Simula.SC.648,
	title = {Characterization of Cyclical CSF Flow in the Foramen Magnum and Upper Cervical Spinal Canal With MR Flow Imaging and Computational Fluid Dynamics},
	journal = {American Journal of Neuroradiology},
	volume = {31},
	number = {6},
	year = {2010},
	month = {June-July},
	pages = {997-1002},
	abstract = {SUMMARYCSF flow has been shown to exhibit complex patterns in MR images in both healthy subjects and in patients with Chiari I. Abnormal CSF flow oscillations, according to prevailing opinion, cause syringomyelia and other clinical manifestations that affect some patients with the Chiari I malformation. For this article, we reviewed the literature on PC MR of CSF flow, collected the published CFD studies relevant to CSF flow, and performed flow simulations. PC MR creates cine and still images of CSF flow and measurements of flow velocities. CFD, a technique used to compute flow and pressure in liquid systems, simulates the CSF flow patterns that occur in a specific geometry or anatomy of the SAS and a specific volume of flow. Published PC MR studies show greater peak CSF velocities and more complex flow patterns in patients with Chiari I than in healthy subjects, with synchronous bidirectional flow one of the characteristic markers of pathologic flow. In mathematic models of the SAS created from high-resolution MR images, CFD displays complex CSF flow patterns similar to those shown in PC MR in patients. CFD shows that the pressure and flow patterns vary from level to level in the upper spinal canal and differ between patients with Chiari and healthy volunteers. In models in which elasticity and motion are incorporated, CFD displays CSF pressure waves in the SAS. PC MR and CFD studies to date demonstrate significant alterations of CSF flow and pressure patterns in patients with Chiari I. CSF flow has nonlaminar complex spatial and temporal variations and associated pressure waves and pressure gradients. Additional simulations of CSF flow supplemented by PC MR will lead to better measures for distinguishing pathologic flow abnormalities that cause syringomyelia, headaches, and other clinical manifestations in Chiari I malformations.},
	doi = {10.3174/ajnr.A1995},
	author = {Hentschel, Susanne and Mardal, Kent-Andre and Alf Emil {L{\o}vgren} and Linge, Svein and Haughton, Victor}
}
@misc {Simula.SC.434,
	title = {Cerebrospinal Fluid Flow Investigations With Modelling and Simulation},
	howpublished = {invited talk at The Chiari Institute, 14th of January, New York, U.S.A.},
	year = {2009},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@article {Simula.SC.522,
	title = {CSF Flow Dynamics at the Cranio-Vertebral Junction Studied With an Idealized Model of the Subarachnoid Space and Computational Flow Analysis},
	journal = {American Journal of Neuroradiology},
	volume = {31},
	number = {1},
	year = {2009},
	pages = {185-192},
	author = {Linge, Svein and Haughton, Victor and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Hans Petter {Langtangen}}
}
@misc {Simula.sc.782,
	title = {CSF Flow Modelling},
	howpublished = {Invited talk at CSR Flow Research Conference at University of Wisconsin, Nov 9},
	year = {2009},
	author = {Mardal, Kent-Andre and Haughton, Victor and Hentschel, Susanne and Hans Petter {Langtangen} and Linge, Svein and Alf Emil {L{\o}vgren} and Valen-Sendstad, Kristian}
}
@inproceedings {Simula.SC.606,
	title = {Effect of Tonsilar Position on Cerebrospinal Fluid Flow in the Spinal Subarachnoid Space Studied With Computational Fluid Dynamics},
	journal = {ASNR - 47th Annual Meeting of the American Society of Neuroradiology},
	year = {2009},
	type = {Conference},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@article {Simula.SC.74,
	title = {Global C{^1} Maps on General Domains},
	journal = {Mathematical Models and Methods in Applied Sciences (M3AS)},
	volume = {19},
	number = {5},
	year = {2009},
	note = {{\textcopyright} Copyright World Scientific Publishing Company},
	pages = {803-832},
	abstract = {In many contexts, there is a need to construct C{^1} maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C{^1}. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are dis- cussed and compared: the transfinite interpolation proposed by Gordon and Hall, the {\textquoteright}pseudo-harmonic{\textquoteright} extension proposed by Gordon and Wixom, a new generalization of the Gordon-Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean value extension proposed by Floater.},
	doi = {10.1142/S0218202509003632},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@misc {Simula.sc.780,
	title = {Patient-Specific Simulations of Stroke and Syringomyelia},
	howpublished = {Talk : Advisory commitee meeting FFI project P1112 Aerosols: Dispersion, Transport and Consequences, Nov 03},
	year = {2009},
	author = {Mardal, Kent-Andre and Hentschel, Susanne and Helgeland, Anders and Hans Petter {Langtangen} and Linge, Svein and Alf Emil {L{\o}vgren} and Logg, Anders and Valen-Sendstad, Kristian}
}
@misc {Simula.SC.440,
	title = {Real-Time Computation of CSF Flow},
	howpublished = {Invited talk at a seminar for cerebrospinal fluid flow, University of Wisconsin, Madison, U.S.A., January 16th},
	year = {2009},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist} and Deparis, Simone}
}
@misc {Simula.SC.438,
	title = {Real-Time Flow Simulation},
	howpublished = {Invited talk at The Chiari Institute, New York, U.S.A., January 14th},
	year = {2009},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist} and Deparis, Simone}
}
@misc {Simula.SC.622,
	title = {The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case},
	howpublished = {Invited talk at the reduced basis mini-symposium at ICOSAHOM 09, Trondheim, Norway, June 22-26},
	year = {2009},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@misc {Simula.SC.436,
	title = {Simulating Normal and Abnormal CSF Flow With Idealized Geometries},
	howpublished = {talk, seminar on cerebrospinal fluid flow, 16th of January, University of Wisconsin, U.S.A.},
	year = {2009},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@misc {Simula.SC.636,
	title = {Some Biomedical Applications at Simula},
	howpublished = {Talk at the CBC Workshop on Computational Biology with UMB},
	year = {2009},
	author = {Mardal, Kent-Andre and Helgeland, Anders and Hentschel, Susanne and Hans Petter {Langtangen} and Logg, Anders and Linge, Svein and Alf Emil {L{\o}vgren} and Valen-Sendstad, Kristian}
}
@misc {Simula.SC.624,
	title = {The Spectral Element Method Used to Assess the Quality of a Deformed Mesh},
	howpublished = {contributed talk at ICOSAHOM 09, Trondheim, Norway, June 22-26},
	year = {2009},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@misc {Simula.SC.646,
	title = {Stabilized Reduced Basis Approximation of the Navier-Stokes Equations in Deformed Domains},
	howpublished = {Contributed talk at the workshop on Model Reduction of Parametrized Systems, M{\"u}nster, Germany, Sept. 16-18},
	year = {2009},
	author = {Alf Emil {L{\o}vgren} and Deparis, Simone}
}
@misc {Simula.SC.590,
	title = {Tonsilar Herniation - How Is the CSF Flow Influenced?},
	howpublished = {Invited talk at Rikshospitalet University Hospital, Oslo, Norway, 5th of June},
	year = {2009},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@misc {Simula.SC.256,
	title = {Cerebrospinal Fluid Flow in Normal and Chiari Subjects},
	howpublished = {talk, Seminar at CBC (Simula Research Laboratory), 16th of December, Oslo, Norway},
	year = {2008},
	month = {December},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@misc {Simula.SC.258,
	title = {Cerebrospinal Fluid Flow Simulations - Normal Subjects and Patients With Chiari I Malformation},
	howpublished = {talk, Seminar at CBC (Simula Research Laboratory), 31st of October, Oslo, Norway)},
	year = {2008},
	month = {October},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@misc {Simula.SC.140,
	title = {CFD Analysis of Cerebrospinal Fluid Flow in the Cranio-Cervical Region},
	howpublished = {Invited talk at the 21st Nordic Seminar on Computational Mechanics, Trondheim, October 16-17},
	year = {2008},
	author = {Alf Emil {L{\o}vgren} and Linge, Svein and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@inproceedings {Simula.SC.144,
	title = {CFD Analysis of Cerebrospinal Fluid Flow in the Cranio-Cervical Region},
	journal = {21st Nordic Seminar on Computational Mechanics},
	year = {2008},
	pages = {94-97},
	publisher = {CIMNE},
	type = {Conference},
	isbn = {978-84-96736-56-6},
	author = {Alf Emil {L{\o}vgren} and Linge, Svein and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}},
	editor = {Kvamsdal, T. and Mathisen, K.M. and Pettersen, B.}
}
@misc {Simula.SC.246,
	title = {Chiari Malformation - the Problem and a Mathematical Approach},
	howpublished = {Invited talk, Vindern Medical Center, 8th of December, Oslo, Norway},
	year = {2008},
	month = {December},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Haughton, Victor and Hans Petter {Langtangen}}
}
@misc {Simula.SC.260,
	title = {Simulating the Cerebral Fluid Flow Associated With Chiari I Malformation in Idealized Geometries},
	howpublished = {talk, Seminar at CBC (Simula Research Laboratory), 25th of June, Oslo, Norway.},
	year = {2008},
	month = {June},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Hans Petter {Langtangen}}
}
@misc {Simula.SC.262,
	title = {Simulating the Cerebral Fluid Flow Associated With Chiari I Malformation in Idealized Geometries},
	howpublished = {talk, Seminar at CBC (Simula Research Laboratory), 20th of May, Oslo, Norway.},
	year = {2008},
	month = {May},
	author = {Linge, Svein and Alf Emil {L{\o}vgren} and Mardal, Kent-Andre and Hans Petter {Langtangen}}
}
@techreport {Simula.SC.25,
	title = {On the Implementation of {\textquoteright}pseudo-Harmonic{\textquoteright} Extension.},
	number = {2/07},
	year = {2007},
	publisher = {Norwegian University of Science and Technology},
	address = {7491 Trondheim},
	abstract = {The \&$\#$39;pseudo-harmonic\&$\#$39; extension is an approximation to the common harmonic extension for extending a function over a domain based on its trace along the boundary of the domain. On a circle the two extension methods produce identical results. We present explicit formulas for the computation of distance functions and intersection points needed in the \&$\#$39;pseudo-harmonic\&$\#$39; extension on a circle, a square, and a pentagon. While the harmonic extension needs the solution of a Laplace problem for each new boundary function, the \&$\#$39;pseudo-harmonic\&$\#$39; extension can reuse the distance functions and intersection points for any piecewise continuous function defined on the boundary of the domain.},
	author = {Alf Emil {L{\o}vgren}}
}
@inbook {Simula.SC.26,
	title = {The Reduced Basis Element Method for Fluid Flows},
	booktitle = {Analysis and Simulation of Fluid Dynamics},
	series = {Advances in Mathematical Fluid Mechanics},
	year = {2007},
	pages = {129-154},
	publisher = {Birkh{\"a}user},
	organization = {Birkh{\"a}user},
	chapter = {8},
	address = {Basel},
	abstract = {The reduced basis element approximation is a discretization method for solving partial differential equations that has inherited features from the domain decomposition method and the reduced basis approximation paradigm in a similar way as the spectral element method has inherited features from domain decomposition methods and spectral approximations. We present here a review of the method directed to the application of fluid flow simulations in hierarchical geometries. We present the rational and the basics of the method together with details on the implementation. We illustrate also the rapid convergence with numerical results.},
	isbn = {978-3-7643-7741-0},
	author = {Alf Emil {L{\o}vgren} and Einar M. {R{\o}nquist} and Maday, Yvon},
	editor = {Calgaro, C. and Coulombel, J.-F. and Goudon, T.}
}
@misc {Simula.SC.130,
	title = {Reduced Basis Modeling of Complex Flow Systems},
	howpublished = {Invited talk at the Modelling and Scientific Computing group at EPFL, Switzerland, November, 2007. Also presented at the workshop on Modelling and Computation of Biomedical Processes at CBC, Simula, June 7-14},
	year = {2007},
	author = {Alf Emil {L{\o}vgren} and Einar M. {R{\o}nquist} and Maday, Yvon}
}
@inproceedings {Lovgren.2006.4,
	title = {A Reduced Basis Element Method for Complex Flow Systems},
	journal = {ECCOMAS CFD 2006, European Conference on Computational Fluid Dynamics},
	year = {2006},
	publisher = {TU Delft},
	type = {Conference},
	abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations within domains belonging to a certain class. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into smaller blocks that are topologically similar to a few reference shapes (or generic computational parts). Associated with each reference shape are precomputed solutions corresponding to the same governing partial differential equation, and similar boundary conditions, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to the computational domain is then taken to be a linear combination of the precomputed solutions, mapped from the reference shapes for the different blocks to the actual domain. The variation of the geometry induces non-affine parameter dependence, and we apply the empirical interpolation technique to achieve an offline/online decoupling of the reduced basis procedure. Some results for incompressible flow systems have already been presented, and the focus here will be to further improve the offline/online decoupling of problems with non-affine parameter dependence. To this end we use the empirical interpolation method to approximate the parameter depen- dent operators. We also present a generalized transfinite interpolation method intended to produce global C1 mappings from the reference shapes to each corresponding block of the computational domain.},
	isbn = {90-9020970-0},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}},
	editor = {Wesseling, P. and O{\~n}ate, E. and P{\'e}riaux, J.}
}
@article {Lovgren.2006.1,
	title = {A Reduced Basis Element Method for the Steady Stokes Problem},
	journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
	volume = {40},
	number = {3},
	year = {2006},
	pages = {529-552},
	abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual computational part. We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition, to a posteriori error estimation, and to {\textquotedblleft}gluing{\textquotedblright} the local solutions together in the multidomain case.},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@article {Lovgren.2006.3,
	title = {A Reduced Basis Element Method for the Steady Stokes Problem: Application to Hierarchical Flow Systems},
	journal = {Model. Identif. Control},
	volume = {27},
	number = {2},
	year = {2006},
	pages = {79-94},
	abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations, and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of sub-domains that are similar to a few reference domains or generic computational parts. Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to the mapping of the velocity fields, to satisfying the inf-sup condition, and to gluing the local solutions together in the multi domain case.We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@inproceedings {Lovgren.2005.2,
	title = {A Reduced Basis Element Method for the Steady Stokes Problem: Application to Hierarchical Flow Systems},
	journal = {SIMS 2005, 46th Conference on Simulation and Modeling},
	year = {2005},
	publisher = {SIMS 2005 and Tapir Academic Press},
	type = {1},
	address = {Trondheim, Norway},
	abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations, and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of sub-domains that are similar to a few reference domains or generic computational parts. Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to the mapping of the velocity fields, to satisfying the inf-sup condition, and to gluing the local solutions together in the multi domain case.We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.},
	author = {Alf Emil {L{\o}vgren} and Maday, Yvon and Einar M. {R{\o}nquist}}
}
@phdthesis {Lovgren.2005.1,
	title = {Reduced Basis Modeling of Hierarchical Flow Systems},
	year = {2005},
	school = {Norwegian University of Science and Technology},
	type = {phd},
	author = {Alf Emil {L{\o}vgren}}
}
