Computer simulation of multiscale problems
|
|
- Andrea Alexander
- 5 years ago
- Views:
Transcription
1 Progress in the SSF project CutFEM, Geometry, and Optimal design Computer simulation of multiscale problems Axel Målqvist and Daniel Elfverson University of Gothenburg and Uppsala University Umeå Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
2 Research group and Outline My research group consists of: Daniel Elfverson, Uppsala, Multiscale and UQ, Fredrik Hellman, Uppsala, Multiscale and UQ, Anna Persson, Göteborg, Time dep. ms problems, Gustav Kettil, Göteborg, Simulation of paper, Tony Stillfjord, Göteborg, Time dep. PDE and splitting, Support from SSF project Introduction to Localized Orthogonal Decomposition techniques for solving multiscale problems (Axel) Application to eigenvalue problems (Axel) Application to problems posed on complex geometry (Daniel) Recent development and openings for collaboration (Axel) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
3 Multiscale problems Applications such as flow in a porous medium composite materials require numerical solution of partial differential equations with rough data (e.g. permeability or module of elasticity). Major challenge: Features on multiple scales in space and time. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
4 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 A(x) x oscillatory coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
5 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
6 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
7 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
8 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
9 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
10 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
11 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
12 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
13 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
14 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
15 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
16 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Conclusion Fine scale features have to be resolved even to get coarse solution behavior right (both H 1 and L 2 errors are large). Resolution of the fine scales by a uniform mesh is very computationally expensive. Local mesh refinement is not an option. The standard basis does not seem to be suitable for this problem. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
17 Standard FE decomposition Coarse FE mesh with parameter H Piecewise linear continuous FE space V H I T : V V H interpolation operator Decomposition V = V H V f with V f := {v V I T v = 0} Example: rough coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
18 Standard FE decomposition Coarse FE mesh with parameter H Piecewise linear continuous FE space V H I T : V V H interpolation operator Decomposition V = V H V f with V f := {v V I T v = 0} Example: = + u V u H V H u f V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
19 Orthogonal multiscale decomposition Let dim(v ms H ) = dim(v H) Start from V H and add fine scale corrections in V f v ms H Decomposition V ms, v H f V f holds Ω A vms v H f dx = 0 V = V ms H V f with V f := {v V I T v = 0} Example: rough coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
20 Orthogonal multiscale decomposition Let dim(v ms H ) = dim(v H) Start from V H and add fine scale corrections in V f v ms H Decomposition V ms, v H f V f holds Ω A vms v H f dx = 0 V = V ms H V f with V f := {v V I T v = 0} Example: = + u V u ms H V ms H u f V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
21 Computing a basis Let V H = span {λ x x N} and further let, φ x V f solve A (λ x φ x ) w dx = 0, for all w V f. Ω Multiscale FE space V ms H = span {λ x φ x x N} Example: = λ x φ x V ms H λ x V H φ x V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
22 Computing a basis Let V H = span {λ x x N} and further let, φ x V f solve A (λ x φ x ) w dx = 0, for all w V f. Ω Multiscale FE space V ms H = span {λ x φ x x N} Example: = λ x φ x V ms H λ x V H φ x V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
23 Computing a basis We have proven exponential decay of λ x φ x! Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
24 Computing a basis This allows us to truncate to a patch. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
25 Computing a basis This allows us to truncate to a patch and fine scale discretization. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
26 Computing a basis Localized patch with refined mesh gives computable basis functions The multiscale basis functions are computed by solving the PDE on local patches. The basis functions are totally independent, leading to trivial parallelization. Exponential decay has been proven and is crucial. Remember dim(v ms) = dim(v H H). Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
27 Computing the multiscale approximation Multiscale approximation: u ms H A u ms H Ω v dx = V ms H Ω satisfies f v dx for all v V ms H We have proven error bound (using k = log(1/h) layers): (u u ms H ) CH f, where v 2 = Ω v2 dx and C is independent on variations in A. Note that for the standard FEM with A = A( x ) we have, ɛ (u u H ) C H ɛ f. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
28 Numer. exp. (Poisson, f L 2 (H), f H 1 0 (H2 )!) H = 2 1, 2 2,..., 2 7 h = 2 9, k = log(1/h) (u h u ms,h ) vs. #dof H,k Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
29 Numer. exp. (linear eigenvalue problem, H 4 ) l λ (l) h e (l) (1/2 2) e (l) (1/4 2) e (l) (1/8 2) e (l) (1/16 2) Table: Errors e (l) (H) =: λ(l) H λ(l) h λ (l) h and h = Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
30 The quadratic eigenvalue problem Consider discretized structure with damping, Kx + λcx + λ 2 Mx = 0, where K ij = (A φ j, φ i ) is stiffness, C ij = c(φ j, φ i ) is damping, and M ij = (φ j, φ i ) is mass matrix. Linearization: (y = λx) ] ] Ax := [ K 0 0 M ] [ x y [ C M = λ M 0 ] [ x y := λbx, Note that T = A 1 B has eigenvalues λ 1. T is not symmetric (even if C is), it is compact but not necessarily in the limit h 0. The analysis is very different from symmetric case. Semi-simple eigenvalues are ok but defective leads to unknown constants. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
31 Numerical experiment H H = 2 1, 2 2,..., 2 5 h = 2 6, k = λ h λ ms / λ H h vs. #dof A (pic), c(u, v) = (1 + sin(10x))u v dx Ω Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
32 Numerical experiment H 2 H = 2 1, 2 2,..., 2 5 h = 2 6, k = 10 3 λ h λ ms / λ H h vs. #dof πc(u, v) = (atan(10x 5) + π )A u v dx Ω 2 Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
33 Relevant papers Some papers relevant for the SSF project (work package D): Poisson P1, M. & Peterseim, Math. Comp., Poisson DG, E., Georgoulis, M., & Petersiem, SINUM, Linear Eig., M. & Peterseim, Numer. Math Gross-Pitaevskii, Henning, M., & Peterseim, SINUM, Helmholtz, Peterseim, two preprints LOD is applicable when: The diffusion is present and lower order terms are not dominating. LOD is efficient when: basis can be reused many times in the calculation (nonlinear iteration, time stepping,...). Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
34 LOD complex geometry Consider the Poisons equation u = f in Ω, ν u = κu on Γ R, u = 0 on Γ D, where Γ D and Γ R are not resolved by the mesh Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
35 Objective Construct a multiscale method for complicated geometries with the following properties: No pre-asymptotic effect Linear convergence without resolving the boundary Conditioning is independent of how the boundary cut the mesh Only enrich the coarse finite element space in the vicinity of the boundary Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
36 Poincaré-Friedrich inequality Difficulties: We need a inequality of the type where inf c R u c L 2 (ω) C(ω)diam(ω) u L 2 (ω) C(ω) is bounded independent of diam(ω) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
37 Error analysis Let ω Γ be a vicinity of the complicated boundary Theorem (Locally enriched LOD method) Given that u H 1 Γ D H 2 (Ω \ ω Γ ) and that u Γ H V Γ,L H solution, then is the LOD u u LOD H h (u I h u) L 2 (ω Γ ) + H u H 2 (Ω\ω Γ ) + H f L 2 (ω Γ ) + H 1 (L) d/2 γ L f L 2 (Ω) holds. The condition number scale as κ H 2. I h u: interpolation operator on the fine mesh L: Number of layers 0 < γ < 1: depends on the Poinaré-Friedrich constant Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
38 Error analysis Choose L = C log(h 1 ), Theorem (Locally enriched LOD method) Given that u H 1 Γ D H 2 (Ω \ ω k 1 Γ ) and u Γ H V Γ,L, then u u LOD H h H holds. The condition number scale as κ H 2. H Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
39 Numerical verification Fractal domain Consider the Poisson equation in the domain: Correctors are computed everywhere Mixed boundary condition Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
40 Numerical verification Fractal domain The convergence in relative energy norm Conditioning Error Fractal O(H) Mesh size H Condition number 10 4 Fractal 10 3 O(H 2 ) Mesh size H Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
41 Numerical verification Local singularities Consider the Poisson equation in the domains: Correctors are computed in the red areas Dirichlet boundary condition Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
42 Numerical verification Local singularities The convergence in relative energy norm Conditioning Error L-shaped Slit O(H) Mesh size H Condition number Mesh size H L-shaped Slit O(H 2 ) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
43 Numerical verification Saw tooth boundary Consider the Poisson equation in the domain: Correctors are computed in the red area Test both Dirichlet and Neumann boundary condition on the saw tooth Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
44 Numerical verification Saw tooth boundary The convergence in relative energy norm Conditioning Error Dirichlet Neumann O(H) Mesh size H Condition number Mesh size H Dirichlet Neumann O(H 2 ) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
45 Numerical verification Cut domain Consider the Poisson equation in the domain: a b c 2) c b a 1) Correctors are computed everywhere Conside a fix mesh and cut the domain in different ways Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
46 Numerical verification Cut domain Cut 1 e rel (a) e rel (b) e rel (c) D D-N N-D cond(a) cond(b) cond(c) D D-N N-D Cut 2 e rel (a) e rel (b) e rel (c) D D-N N-D cond(a) cond(b) cond(c) D D-N N-D Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
47 Recent development and collaboration We consider a Poisson type model problem with a diffusion coefficient A that varies between two different values: A u = f in Ω, u = 0 on Ω. The objects may be short fibres/particles or cracks/holes the solution has low regularity at the boundary of the objects local mesh refinement work for single configuration but not when multiple configurations or moving objects are allowed Monte Carlo simulation or optimization procedure Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
48 Recent development In order to treat multiple configurations we want a regular static coarse background mesh. We only compute LOD basis functions in a region around each object and use standard P1 FEM basis functions in between. The size of the surrounding region can be controlled by H to meet tolerance, p adaptivity, DG.. If the regions overlap we compute basis functions for the union of the overlapping domains. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
49 Recent development CutFEM allows us (we hope) to glue the LOD computations to the background mesh avoiding remeshing. The assembly of the LOD parts is independent of the orientation of the object (rotation,translation). If the objects have the same shape huge savings are possible. Compared to CutFEM using spatially adapted subgrids around the objects the LOD approach leads to much smaller systems of equations for each configuration of the objects. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
50 Recent development Random distribution of objects: Monte-Carlo simulation can be performed on the coarse scale with fine scale accuracy. Design of materials: Optimization procedure where the objects are moved in order to minimize some output. Again repeated solves on the coarse scale still gives fine scale accuracy. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
51 Comments/questions/possible collaborations The problem is very similar to complex boundary, holes is a direct application. Any comments from or difficulties seen by the CutFEM experts? We got the idea from porous media flow applications, where there is a great uncertainty in the location and direction of defects. We seek applications in composite materials which fits our framework. Uncertainty quantification in material science is a possible application for this approach. What about applicability to optimal design problems? We will start this work after the summer, in collaboration with Fredrik Hellman Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30
arxiv: v2 [math.na] 31 Oct 2016
1 Multiscale methods for problems with comple geometry Daniel Elfverson, Mats G. Larson, and Ael Målqvist September 10, 2018 arxiv:1509.0991v2 [math.na] 1 Oct 2016 Abstract We propose a multiscale method
More informationMultiscale methods for time-harmonic acoustic and elastic wave propagation
Multiscale methods for time-harmonic acoustic and elastic wave propagation Dietmar Gallistl (joint work with D. Brown and D. Peterseim) Institut für Angewandte und Numerische Mathematik Karlsruher Institut
More informationNumerical Analysis of Evolution Problems in Multiphysics
Thesis for the Degree of Doctor of Philosophy Numerical Analysis of Evolution Problems in Multiphysics Anna Persson Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology
More informationNonparametric density estimation for elliptic problems with random perturbations
Nonparametric density estimation for elliptic problems with random perturbations, DqF Workshop, Stockholm, Sweden, 28--2 p. /2 Nonparametric density estimation for elliptic problems with random perturbations
More informationA generalized finite element method for linear thermoelasticity
Thesis for the Degree of Licentiate of Philosophy A generalized finite element method for linear thermoelasticity Anna Persson Division of Mathematics Department of Mathematical Sciences Chalmers University
More informationReduction of Finite Element Models of Complex Mechanical Components
Reduction of Finite Element Models of Complex Mechanical Components Håkan Jakobsson Research Assistant hakan.jakobsson@math.umu.se Mats G. Larson Professor Applied Mathematics mats.larson@math.umu.se Department
More informationSpectral element agglomerate AMGe
Spectral element agglomerate AMGe T. Chartier 1, R. Falgout 2, V. E. Henson 2, J. E. Jones 4, T. A. Manteuffel 3, S. F. McCormick 3, J. W. Ruge 3, and P. S. Vassilevski 2 1 Department of Mathematics, Davidson
More informationMultigrid absolute value preconditioning
Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationRemarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable?
Remarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable? Thomas Apel, Hans-G. Roos 22.7.2008 Abstract In the first part of the paper we discuss minimal
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationINVESTIGATION OF STABILITY AND ACCURACY OF HIGH ORDER GENERALIZED FINITE ELEMENT METHODS HAOYANG LI THESIS
c 2014 Haoyang Li INVESTIGATION OF STABILITY AND ACCURACY OF HIGH ORDER GENERALIZED FINITE ELEMENT METHODS BY HAOYANG LI THESIS Submitted in partial fulfillment of the requirements for the degree of Master
More informationNumerische Mathematik
Numer. Math. 2018) 138:191 217 https://doi.org/10.1007/s00211-017-0905-7 Numerische Mathemati Multiscale techniques for parabolic equations Axel Målqvist 1 Anna Persson 1 Received: 30 April 2015 / Revised:
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationAspects of Multigrid
Aspects of Multigrid Kees Oosterlee 1,2 1 Delft University of Technology, Delft. 2 CWI, Center for Mathematics and Computer Science, Amsterdam, SIAM Chapter Workshop Day, May 30th 2018 C.W.Oosterlee (CWI)
More informationLehrstuhl Informatik V. Lehrstuhl Informatik V. 1. solve weak form of PDE to reduce regularity properties. Lehrstuhl Informatik V
Part I: Introduction to Finite Element Methods Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Necel Winter 4/5 The Model Problem FEM Main Ingredients Wea Forms and Wea
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationKinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles
Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. The 8th International Conference on Computational
More informationFEM Convergence for PDEs with Point Sources in 2-D and 3-D
FEM Convergence for PDEs with Point Sources in 2-D and 3-D Kourosh M. Kalayeh 1, Jonathan S. Graf 2 Matthias K. Gobbert 2 1 Department of Mechanical Engineering 2 Department of Mathematics and Statistics
More informationNumerical Solution I
Numerical Solution I Stationary Flow R. Kornhuber (FU Berlin) Summerschool Modelling of mass and energy transport in porous media with practical applications October 8-12, 2018 Schedule Classical Solutions
More informationThe Virtual Element Method: an introduction with focus on fluid flows
The Virtual Element Method: an introduction with focus on fluid flows L. Beirão da Veiga Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca VEM people (or group representatives): P.
More informationModel reduction for multiscale problems
Model reduction for multiscale problems Mario Ohlberger Dec. 12-16, 2011 RICAM, Linz , > Outline Motivation: Multi-Scale and Multi-Physics Problems Model Reduction: The Reduced Basis Approach A new Reduced
More informationModelling in photonic crystal structures
Modelling in photonic crystal structures Kersten Schmidt MATHEON Nachwuchsgruppe Multiscale Modelling and Scientific Computing with PDEs in collaboration with Dirk Klindworth (MATHEON, TU Berlin) Holger
More informationCIMPA Summer School on Current Research in Finite Element Methods
CIMPA Summer School on Current Research in Finite Element Methods Local zooming techniques for the finite element method Alexei Lozinski Laboratoire de mathématiques de Besançon Université de Franche-Comté,
More informationarxiv: v3 [math.na] 10 Dec 2015
arxiv:1410.0293v3 [math.na] 10 Dec 2015 Localized Harmonic Characteristic Basis Functions for Multiscale Finite Element Methods Leonardo A. Poveda Sebastian Huepo Juan Galvis Contents Victor M. Calo October
More informationLocal Mesh Refinement with the PCD Method
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 1, pp. 125 136 (2013) http://campus.mst.edu/adsa Local Mesh Refinement with the PCD Method Ahmed Tahiri Université Med Premier
More informationSolving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department
More informationConstrained Minimization and Multigrid
Constrained Minimization and Multigrid C. Gräser (FU Berlin), R. Kornhuber (FU Berlin), and O. Sander (FU Berlin) Workshop on PDE Constrained Optimization Hamburg, March 27-29, 2008 Matheon Outline Successive
More informationOptimal Left and Right Additive Schwarz Preconditioning for Minimal Residual Methods with Euclidean and Energy Norms
Optimal Left and Right Additive Schwarz Preconditioning for Minimal Residual Methods with Euclidean and Energy Norms Marcus Sarkis Worcester Polytechnic Inst., Mass. and IMPA, Rio de Janeiro and Daniel
More informationSolving PDEs with Multigrid Methods p.1
Solving PDEs with Multigrid Methods Scott MacLachlan maclachl@colorado.edu Department of Applied Mathematics, University of Colorado at Boulder Solving PDEs with Multigrid Methods p.1 Support and Collaboration
More informationarxiv: v1 [math.na] 14 Jun 2017
MULTISCALE DIFFERENTIAL RICCATI EQUATIONS FOR LINEAR QUADRATIC REGULATOR PROBLEMS arxiv:176.438v1 [math.na] 14 Jun 217 AXEL MÅLQVIST, ANNA PERSSON, AND TONY STILLFJORD Abstract. We consider approximations
More informationImplementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs
Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation
More informationChapter 1: The Finite Element Method
Chapter 1: The Finite Element Method Michael Hanke Read: Strang, p 428 436 A Model Problem Mathematical Models, Analysis and Simulation, Part Applications: u = fx), < x < 1 u) = u1) = D) axial deformation
More informationShort title: Total FETI. Corresponding author: Zdenek Dostal, VŠB-Technical University of Ostrava, 17 listopadu 15, CZ Ostrava, Czech Republic
Short title: Total FETI Corresponding author: Zdenek Dostal, VŠB-Technical University of Ostrava, 17 listopadu 15, CZ-70833 Ostrava, Czech Republic mail: zdenek.dostal@vsb.cz fax +420 596 919 597 phone
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationRobust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms
www.oeaw.ac.at Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms Y. Efendiev, J. Galvis, R. Lazarov, J. Willems RICAM-Report 2011-05 www.ricam.oeaw.ac.at
More informationSuitability of LPS for Laminar and Turbulent Flow
Suitability of LPS for Laminar and Turbulent Flow Daniel Arndt Helene Dallmann Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics VMS 2015 10th International Workshop on
More informationFast Structured Spectral Methods
Spectral methods HSS structures Fast algorithms Conclusion Fast Structured Spectral Methods Yingwei Wang Department of Mathematics, Purdue University Joint work with Prof Jie Shen and Prof Jianlin Xia
More informationFinal Ph.D. Progress Report. Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo
Final Ph.D. Progress Report Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo Dissertation Committee: I. Babuska, L. Demkowicz, C. Torres-Verdin, R. Van
More informationRelaxing the CFL condition for the wave equation on adaptive meshes
Wegelerstraße 6 53115 Bonn Germany phone +49 228 73-3427 fax +49 228 73-7527 www.ins.uni-bonn.de D. Peterseim, M. Schedensack Relaxing the CFL condition for the wave equation on adaptive meshes INS Preprint
More informationMyopic Models of Population Dynamics on Infinite Networks
Myopic Models of Population Dynamics on Infinite Networks Robert Carlson Department of Mathematics University of Colorado at Colorado Springs rcarlson@uccs.edu June 30, 2014 Outline Reaction-diffusion
More informationA very short introduction to the Finite Element Method
A very short introduction to the Finite Element Method Till Mathis Wagner Technical University of Munich JASS 2004, St Petersburg May 4, 2004 1 Introduction This is a short introduction to the finite element
More informationAn asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling
An asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3 Saint-Malo 13 December 2016 1 Université
More informationOn Nonlinear Dirichlet Neumann Algorithms for Jumping Nonlinearities
On Nonlinear Dirichlet Neumann Algorithms for Jumping Nonlinearities Heiko Berninger, Ralf Kornhuber, and Oliver Sander FU Berlin, FB Mathematik und Informatik (http://www.math.fu-berlin.de/rd/we-02/numerik/)
More informationGeneralized Finite Element Methods for Three Dimensional Structural Mechanics Problems. C. A. Duarte. I. Babuška and J. T. Oden
Generalized Finite Element Methods for Three Dimensional Structural Mechanics Problems C. A. Duarte COMCO, Inc., 7800 Shoal Creek Blvd. Suite 290E Austin, Texas, 78757, USA I. Babuška and J. T. Oden TICAM,
More informationFEM Convergence for PDEs with Point Sources in 2-D and 3-D
FEM Convergence for PDEs with Point Sources in -D and 3-D Kourosh M. Kalayeh 1, Jonathan S. Graf, and Matthias K. Gobbert 1 Department of Mechanical Engineering, University of Maryland, Baltimore County
More informationASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR THE POINTWISE GRADIENT ERROR ON EACH ELEMENT IN IRREGULAR MESHES. PART II: THE PIECEWISE LINEAR CASE
MATEMATICS OF COMPUTATION Volume 73, Number 246, Pages 517 523 S 0025-5718(0301570-9 Article electronically published on June 17, 2003 ASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR TE POINTWISE GRADIENT
More informationR T (u H )v + (2.1) J S (u H )v v V, T (2.2) (2.3) H S J S (u H ) 2 L 2 (S). S T
2 R.H. NOCHETTO 2. Lecture 2. Adaptivity I: Design and Convergence of AFEM tarting with a conforming mesh T H, the adaptive procedure AFEM consists of loops of the form OLVE ETIMATE MARK REFINE to produce
More informationSparse Linear Systems. Iterative Methods for Sparse Linear Systems. Motivation for Studying Sparse Linear Systems. Partial Differential Equations
Sparse Linear Systems Iterative Methods for Sparse Linear Systems Matrix Computations and Applications, Lecture C11 Fredrik Bengzon, Robert Söderlund We consider the problem of solving the linear system
More informationAdaptive methods for control problems with finite-dimensional control space
Adaptive methods for control problems with finite-dimensional control space Saheed Akindeinde and Daniel Wachsmuth Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy
More informationNumerical Analysis and Methods for PDE I
Numerical Analysis and Methods for PDE I A. J. Meir Department of Mathematics and Statistics Auburn University US-Africa Advanced Study Institute on Analysis, Dynamical Systems, and Mathematical Modeling
More informationMaximum norm estimates for energy-corrected finite element method
Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de,
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation
More informationA posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem
A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem Larisa Beilina Michael V. Klibanov December 18, 29 Abstract
More informationAcceleration of a Domain Decomposition Method for Advection-Diffusion Problems
Acceleration of a Domain Decomposition Method for Advection-Diffusion Problems Gert Lube 1, Tobias Knopp 2, and Gerd Rapin 2 1 University of Göttingen, Institute of Numerical and Applied Mathematics (http://www.num.math.uni-goettingen.de/lube/)
More informationExponential integrators for semilinear parabolic problems
Exponential integrators for semilinear parabolic problems Marlis Hochbruck Heinrich-Heine University Düsseldorf Germany Innsbruck, October 2004 p. Outline Exponential integrators general class of methods
More informationExperiences with Model Reduction and Interpolation
Experiences with Model Reduction and Interpolation Paul Constantine Stanford University, Sandia National Laboratories Qiqi Wang (MIT) David Gleich (Purdue) Emory University March 7, 2012 Joe Ruthruff (SNL)
More informationNUMERICAL HOMOGENIZATION FOR INDEFINITE H(CURL)-PROBLEMS
Proceedings of EQUADIFF 2017 pp. 137 146 NUMERICAL HOMOGENIZATION FOR INDEFINITE H(CURL)-PROBLEMS BARBARA VERFÜRTH Abstract. In this paper, we present a numerical homogenization scheme for indefinite,
More informationA Framework for Analyzing and Constructing Hierarchical-Type A Posteriori Error Estimators
A Framework for Analyzing and Constructing Hierarchical-Type A Posteriori Error Estimators Jeff Ovall University of Kentucky Mathematics www.math.uky.edu/ jovall jovall@ms.uky.edu Kentucky Applied and
More informationSolving PDEs with freefem++
Solving PDEs with freefem++ Tutorials at Basque Center BCA Olivier Pironneau 1 with Frederic Hecht, LJLL-University of Paris VI 1 March 13, 2011 Do not forget That everything about freefem++ is at www.freefem.org
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationNotes for CS542G (Iterative Solvers for Linear Systems)
Notes for CS542G (Iterative Solvers for Linear Systems) Robert Bridson November 20, 2007 1 The Basics We re now looking at efficient ways to solve the linear system of equations Ax = b where in this course,
More informationFinite Elements. Colin Cotter. January 15, Colin Cotter FEM
Finite Elements January 15, 2018 Why Can solve PDEs on complicated domains. Have flexibility to increase order of accuracy and match the numerics to the physics. has an elegant mathematical formulation
More informationSpline Element Method for Partial Differential Equations
for Partial Differential Equations Department of Mathematical Sciences Northern Illinois University 2009 Multivariate Splines Summer School, Summer 2009 Outline 1 Why multivariate splines for PDEs? Motivation
More informationMath 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework
Math 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework Jan Mandel University of Colorado Denver May 12, 2010 1/20/09: Sec. 1.1, 1.2. Hw 1 due 1/27: problems
More informationCONVERGENCE THEORY. G. ALLAIRE CMAP, Ecole Polytechnique. 1. Maximum principle. 2. Oscillating test function. 3. Two-scale convergence
1 CONVERGENCE THEOR G. ALLAIRE CMAP, Ecole Polytechnique 1. Maximum principle 2. Oscillating test function 3. Two-scale convergence 4. Application to homogenization 5. General theory H-convergence) 6.
More informationLecture 1. Finite difference and finite element methods. Partial differential equations (PDEs) Solving the heat equation numerically
Finite difference and finite element methods Lecture 1 Scope of the course Analysis and implementation of numerical methods for pricing options. Models: Black-Scholes, stochastic volatility, exponential
More informationSchwarz Preconditioner for the Stochastic Finite Element Method
Schwarz Preconditioner for the Stochastic Finite Element Method Waad Subber 1 and Sébastien Loisel 2 Preprint submitted to DD22 conference 1 Introduction The intrusive polynomial chaos approach for uncertainty
More informationAdaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA www.math.umd.edu/ rhn 7th
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationarxiv: v2 [math.na] 28 Nov 2010
Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems Ivo Babuska 1 Robert Lipton 2 arxiv:1004.3041v2 [math.na] 28 Nov 2010 Abstract The paper
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationBOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET
BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET WEILIN LI AND ROBERT S. STRICHARTZ Abstract. We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski
More informationIntroduction to Wavelet Based Numerical Homogenization
Introduction to Wavelet Based Numerical Homogenization Olof Runborg Numerical Analysis, School of Computer Science and Communication, KTH RTG Summer School on Multiscale Modeling and Analysis University
More informationTwo new enriched multiscale coarse spaces for the Additive Average Schwarz method
346 Two new enriched multiscale coarse spaces for the Additive Average Schwarz method Leszek Marcinkowski 1 and Talal Rahman 2 1 Introduction We propose additive Schwarz methods with spectrally enriched
More informationAn inverse problem for a system of steady-state reaction-diffusion equations on a porous domain using a collage-based approach
Journal of Physics: Conference Series PAPER OPEN ACCESS An inverse problem for a system of steady-state reaction-diffusion equations on a porous domain using a collage-based approach To cite this article:
More informationarxiv: v1 [math.na] 25 Aug 2016
A conservative local multiscale model reduction technique for Stoes flows in heterogeneous perforated domains Eric T. Chung Maria Vasilyeva Yating Wang arxiv:1608.07268v1 [math.na] 25 Aug 2016 October
More informationCoupling atomistic and continuum modelling of magnetism
Coupling atomistic and continuum modelling of magnetism M. Poluektov 1,2 G. Kreiss 2 O. Eriksson 3 1 University of Warwick WMG International Institute for Nanocomposites Manufacturing 2 Uppsala University
More informationPartial regularity for fully nonlinear PDE
Partial regularity for fully nonlinear PDE Luis Silvestre University of Chicago Joint work with Scott Armstrong and Charles Smart Outline Introduction Intro Review of fully nonlinear elliptic PDE Our result
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationNONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS. Max Gunzburger
NONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS Max Gunzburger Department of Scientific Computing Florida State University North Carolina State University, March 10, 2011
More informationUpscaling Wave Computations
Upscaling Wave Computations Xin Wang 2010 TRIP Annual Meeting 2 Outline 1 Motivation of Numerical Upscaling 2 Overview of Upscaling Methods 3 Future Plan 3 Wave Equations scalar variable density acoustic
More informationMachine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method
Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method Ivo Babuška Xu Huang Robert Lipton May 28, 2013 Abstract A multiscale spectral generalized finite
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM Repetition of Previous Lecture Sparse grids in Uncertainty Quantification
More informationIn Proc. of the V European Conf. on Computational Fluid Dynamics (ECFD), Preprint
V European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010 J. C. F. Pereira and A. Sequeira (Eds) Lisbon, Portugal, 14 17 June 2010 THE HIGH ORDER FINITE ELEMENT METHOD FOR STEADY CONVECTION-DIFFUSION-REACTION
More informationMultigrid Methods for Saddle Point Problems
Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In
More informationEfficient Solvers for Stochastic Finite Element Saddle Point Problems
Efficient Solvers for Stochastic Finite Element Saddle Point Problems Catherine E. Powell c.powell@manchester.ac.uk School of Mathematics University of Manchester, UK Efficient Solvers for Stochastic Finite
More informationWeak Galerkin Finite Element Methods and Applications
Weak Galerkin Finite Element Methods and Applications Lin Mu mul1@ornl.gov Computational and Applied Mathematics Computationa Science and Mathematics Division Oak Ridge National Laboratory Georgia Institute
More informationAxioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates)
Axioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates) Carsten Carstensen Humboldt-Universität zu Berlin 2018 International Graduate Summer School on Frontiers of Applied and
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationConvergence and optimality of an adaptive FEM for controlling L 2 errors
Convergence and optimality of an adaptive FEM for controlling L 2 errors Alan Demlow (University of Kentucky) joint work with Rob Stevenson (University of Amsterdam) Partially supported by NSF DMS-0713770.
More informationElliptic Equations in High-Contrast Media and Applications
arxiv:1410.1015v1 [math.na] 4 Oct 2014 Elliptic Equations in High-Contrast Media and Applications Ecuaciones Elípticas en Medios de Alto Contraste y Aplicaciones Master Thesis Leonardo Andrés Poveda Cuevas
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationIterative Methods and Multigrid
Iterative Methods and Multigrid Part 1: Introduction to Multigrid 1 12/02/09 MG02.prz Error Smoothing 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Initial Solution=-Error 0 10 20 30 40 50 60 70 80 90 100 DCT:
More informationSUPG Reduced Order Models for Convection-Dominated Convection-Diffusion-Reaction Equations
SUPG Reduced Order Models for Convection-Dominated Convection-Diffusion-Reaction Equations Swetlana Giere a,, Traian Iliescu b,1, Volker John a,c, David Wells b,1, a Weierstrass Institute for Applied Analysis
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationPARTITION OF UNITY FOR THE STOKES PROBLEM ON NONMATCHING GRIDS
PARTITION OF UNITY FOR THE STOES PROBLEM ON NONMATCHING GRIDS CONSTANTIN BACUTA AND JINCHAO XU Abstract. We consider the Stokes Problem on a plane polygonal domain Ω R 2. We propose a finite element method
More information