On Multigrid for Phase Field
|
|
- Baldric Johnson
- 5 years ago
- Views:
Transcription
1 On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004
2 Synopsis adaptivity in time and space fast, robust solution of discrete spatial problems vector-valued Allen Cahn equations (Kh. & Krause) obstacle potential: monotone multigrid (MMG) by successive 1-D minimization logarithmic potential: robustness by constrained Newton linearization numerical experiments diffusion induced grain boundary motion (Kh., Krause & Styles) phase field models multigrid algorithms for the phase field and the solute Cahn Hilliard equation with obstacle potential (Gräser & Kh.) minimization, saddle point formulation and Uzawa-type iteration numerical experiments
3 Ginzburg Landau Approach Ginzburg Landau free energy: E(u) = phase field: u [ 1, +1] Ω ε u ψ(u) dx ε ψ T logarithmic free energy: ψ T (u) = 1 2T((1 u) ln(1 u 2 ) + (1 + u) ln( 1+u 2 )) T c(1 u 2 ) temperature T, critical temperature T c u deep quench limit: T 0 ψ T ψ 0 robustness of numerical solvers for T 0: no classical Newton linearization
4 Phase Field Models isothermal case: T = const. phase transition: Allen-Cahn equation (non-conserving) εu t = d du E(u) = ε u 1 ε ψ (u) (Cahn 60, Allen & Cahn 77) phase separation: Cahn-Hilliard equations (conserving) εu t = w, w = d du E(u) = ε u + 1 ε ψ (u) (Cahn & Hilliard 58) Lyapunov functional: d dt E(u(t)) 0 deep quench limit: complementarity problems, variational inequalities
5 Phase Transition with N Phases concentrations: u1,...,un vector-valued phase field: u = (u1,...,un) R N Gibbs simplex: u(x, t) G = {v R N 0 vi, N i=1 vi = 1} (closed, convex) examples: N=2: u 1 u 2 G N=3: u 1 u 3 u 2 G
6 Vector-valued Allen-Cahn Equation: Obstacle Potential Ginzburg-Landau free energy: E(u) = Ω ε N 2 i=1 u i εψ(u) dx, ε > 0 obstacle potential: ψ = ψ 0 = T c N 2 N i=1 u i(1 u i ) u G parabolic variational inequality (Bronsard & Reitich 93, Garcke et al. 98, 99a, 99b): u(t) G : Gibbs constraints: G = ε(u t, v u) + ε( u, (v u)) T cn ε (u,v u) 0 v {v ( H 1 (Ω) ) N v(x) G a.e. in Ω } G special case N = 2: scalar free energy for u := u 1 u 2
7 Discretization semi-implicit Euler discretization (concave part explicit): stepsize τ (unconditionally stable) linear finite elements S N j : triangulation T j, meshsize h j = O(2 j ), nodes N j, nodal basis λ (j) p discrete spatial problems: u k j G j : u k j, v u k j + τ( u k j, (v u k j)) }{{} a(u k j,v uk j ) τ 1 + T c N ε 2uk 1 j, v u k j }{{} l(v u k j ) k uj G j discrete Gibbs constraints: G j = { v S N j v(p) G p N j }
8 Equivalent Formulation: Convex Minimization u j G j : J (u j ) J (v) v G j quadratic energy: J (v) = 1 2a(v, w) l(v) s.p.d. bilinear form: a(v, w) = v, w + τ( v, w) linear functional: l(v) = 1 + T c N τ u k 1 ε 2 j,v closed convex set: G j = { v S N j v(p) G p N j } Sj iterative solution by descent methods
9 Polygonal Gauß-Seidel Relaxation (Kh. & Krause 02) descent directions: λ (j) p E m, p N j, m = 1,...,M edge vectors of G: E 1,...,E M R N, M := N(N 1) 2 = O(N 2 ) E 2 E 3 G hyperplane: H j = span{λ (j) p E m p N j,m = 1,...,M} E 1 polygonal splitting: H j = p N j M m=1 V (j) p,m, V (j) p,m = span{λ (j) p E m } complexity: O(N 2 n j ) successive minimization of J = J + χ Gj on 1-D subspaces V (j) p,m for p N j and m = 1,...,M
10 convergence of subsequence: u ν k j Convergence u = u pλ (j) p G j u is fixed point: a(u,λ (j) p η i ) l(λ (j) p η i ) η i span{e i }, u p + η i G Lemma (Kh., Krause & Ziegler) u p, v p G there is a decomposition N 1 v p u p = η i, η i span{e mi }, u p + η i G i=1 v p u p u = u j is the solution: a(u,λ (j) p (v p u p)) l(λ (j) p (v p u p)) v p G a(u,v u ) l(v u ) v G j
11 A Variant: Block Gauß-Seidel nodal splitting: H j = p N j V p, V p = {λ (j) p E E span{e 1,...,E M }} successive minimization of J + χ Gj on V p : convergence proof similar (but simpler) solution of (N 1) D subproblems complicated for large N typical drawback of any Gauß-Seidel approach: convergence speed might deteriorate exponentially (i.e. ρ j = 1 O(2 j ))
12 Multilevel Version of Polygonal Gauß-Seidel adaptive refinement: T 0 T 1 T j nodes: N 0 N 1 N j nested finite element spaces: S 0 S 1 S j, λ (j 2) q S k = span{λ (k) p p N k } λ (j) p q p polygonal multilevel splitting: H j = j k=0 p N k M m=1 V (k) p,m V (k) p,m = span{λ (k) p E m }
13 Polygonal Monotone Multigrid (Kh. & Krause 02) sucessive minimization of J = J + χ Gj on subspaces V (k) p,m for p N k, m = 1,...,M and k = j,j 1,...,0 k = j polygonal Gauß-Seidel relaxation on S N j : fine grid smoother: M j (convergent descent method) k < j coarse grid correction: C j (monotone: J (C j w)) J (C j w)) technical modifications (monotone restriction, truncation) monotone multigrid: u ν+1 j = C j ū ν j ū ν j = M ju ν j properties: implementation as multigrid V -cycle: complexity O(N 2 n j ) globally convergent: u ν j u j ν asymptotic multigrid convergence rates for N = 2
14 logarithmic potential: Logarithmic Potential ψ T (u) = Tφ(u) + φ 0 (u) convex part: φ(u) = N i=1 u i ln(u i ) concave part: φ 0 (u) = T c N 2 N i=1 u i(1 u i ) ψ T has N distinct minima for 0 T< T c spatial problems: u j G j : a(u j,v) + τ ε 2 T φ (u j ),v = l(v) v H j minimization of convex energy J T (v) = 1 2 a(v, v) + τ ε 2 T φ(v), 1 l(v): u j G j : J T (u j ) J T (v) v G j family of problems: T > 0 logarithmic potential, T = 0 obstacle potential robustness: family of convergent algorithms T 0
15 What can we do? nonlinear polygonal Gauß-Seidel: robust but slow multilevel version: too expensive (no representation of φ on S k, k < j) Newton s method: not robust (not applicable for T = 0) constrained Newton linearization: small Newton corrections implicit local damping
16 Robust Multigrid for the Logarithmic Potential (Kh. & Krause 04) given iterate u ν j fine grid smoothing: M j (globally convergent descent method) nonlinear polygonal Gauß-Seidel iteration: smoothed iterate ū ν j = M ju ν j coarse grid correction: C j (monotone) Newton linearization of φ at ū ν j constrain corrections to smooth regime G u ν j of φ 1 step of polygonal MMG with local damping G Gūν j (p) ū ν j (p) new iterate u ν+1 j = C j ū ν j family of multigrid algorithms T 0 : V -cycle with complexity O(N 2 n j ), (asymptotic) multigrid convergence
17 Numerical Experiments: The Scalar Case N = 2 time discretization: backward Euler parameter: ε = 0.005, T = 1 10 T c, T c = 1, τ = ε2, j = 7 (h j = 1/256 ε/2) initial condition: u(x, 0) 1, random evolution: t = 100τ t = 500τ t = 2000τ t = 3000τ
18 Robust Convergence parameter: ε = 0.005, T c = 1, τ = 1 2 ε2 initial condition: u(x, 0) 1, random algorithm: multigrid V (1, 1)-cycle with nested iteration, uniform refinement various time steps (T = 0.001, h j = 1/256) meshsize: h j 0 (T = 0.001, 1. time step) temperature T 0 (1. time step, h j = 1/256) time steps k refinement level j inverse temperature 1/T
19 N = 5 Phases in 2D parameter: logarithmic potential ψ T, 0 T T c = 1, ε = 0.08 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /80, linear finite elements: uniform refinement j = 6 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration time step τ (T = 0.0, h j = 1/64) mesh size: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0 h j = 1/128) time step size τ refinement level j inverse temperature 1/T
20 N = 3 Phases in 3D parameter: logarithmic potential ψ 0, T c = 1, ε = 0.16 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /2, linear finite elements: uniform refinement j = 5 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration timesteps: τ k (T = 0.0, h j = 1/32) meshsize: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0, h j = 1/32) time step size τ refinement level j inverse temperature 1/T
21 Conclusion fast, robust solver for arbitrary N and 2 or 3 space dimensions convergence slows down for T > 0 and increasing τ: local damping strategy too pessimistic Current Work anisotropic interfacial energy (cf. Garcke, Nestler & Stoth 99) adaptivity in time and space heuristic refinement indicators, a posteriori error estimates (cf. Kessler, Nochetto & Schmidt 03)
22 Diffusion Induced Grain Boundary Motion (Garcke & Styles 04) un-alloyed and alloyed grains: ϕ i and ϕ i+n, i = 1,...,N order parameter: ϕ = (ϕ 1,...,ϕ 2N ), free energy with obstacle potential: E(ϕ,u) = ( 1 Ω 2ε u2 + ε N 2 i=1 (ϕ i + ϕ i+n ) ψ 0(ϕ) + g(u) ) N i=1 ϕ i dx alloying solute: u semi-linear parabolic equation: u t (D(ϕ) u) = q(u) grain 3 grain 2 degenerate diffusion: D(ϕ) = 2N i=1 (1 ϕ i) grain 1
23 Multigrid for the Phase Field variational inequality: ϕ(t) G : ε(ϕ t, η ϕ) + ε( ϕ, ( ϕ ṽ)) ( 1 εϕ + g(u)e, ϕ ṽ) v G ṽ = (v i + v i+n ) N i=1 RN, e = (1,...,1,0,...,0) R 2N semi-implicit Euler method, finite elements: ϕ k+1 j, v ϕ k+1 j + τ( ϕ k+1 j, (ṽ ϕ k+1 j )) }{{} a(ϕ k+1 j,v ϕ k+1 j ) ( τ + 1)ϕ k ε 2 j + τ ε g(ϕk j )e, v ϕk+1 j }{{} l(v ϕ k+1 j ) fast solution by monotone multigrid complexity: O((2N) 2 n j ), alternative model (Elliott & Styles 03): N phases
24 Multigrid for the Solute semi-implicit Euler method, finite elements: u k+1 j S j : ϕ k+1,v + τ(d(ϕ k j ) uk+1 j, v) = u k j, v + τ(q(uk j ), v) v S j j observation: u k+1 j (p) = u k j (p) p N j = {p N j D(ϕ k j ) suppλ p (j) 0} reduced solution space: S j = {v Ω v S j} Ω = Ω \ Ω, Ω = p N j suppλ (j) p reduced Neumann problem: u k+1 j S j : b(uk+1 j,v) = f(v) v S j no representation of Ω on coarse grids: truncated multigrid (Kh. & Yserentant 94)
25 Current work numerical results comparison of thin/thick film models with 3D realistic geometries improved models effects of continuum mechanics
Constrained 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 informationNonsmooth Schur Newton Methods and Applications
Nonsmooth Schur Newton Methods and Applications C. Gräser, R. Kornhuber, and U. Sack IMA Annual Program Year Workshop Numerical Solutions of Partial Differential Equations: Fast Solution Techniques November
More informationOn preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints
On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints Carsten Gräser and Ralf Kornhuber FU Berlin, FB Mathematik und Informatik (http://www.math.fu-berlin.de/rd/we-02/numerik/)
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationA sharp diffuse interface tracking method for approximating evolving interfaces
A sharp diffuse interface tracking method for approximating evolving interfaces Vanessa Styles and Charlie Elliott University of Sussex Overview Introduction Phase field models Double well and double obstacle
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 informationMultigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids
Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids Long Chen 1, Ricardo H. Nochetto 2, and Chen-Song Zhang 3 1 Department of Mathematics, University of California at Irvine. chenlong@math.uci.edu
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationA Recursive Trust-Region Method for Non-Convex Constrained Minimization
A Recursive Trust-Region Method for Non-Convex Constrained Minimization Christian Groß 1 and Rolf Krause 1 Institute for Numerical Simulation, University of Bonn. {gross,krause}@ins.uni-bonn.de 1 Introduction
More informationLECTURE 3: DISCRETE GRADIENT FLOWS
LECTURE 3: DISCRETE GRADIENT FLOWS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and
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 informationModelling of interfaces and free boundaries
University of Regensburg Regensburg, March 2009 Outline 1 Introduction 2 Obstacle problems 3 Stefan problem 4 Shape optimization Introduction What is a free boundary problem? Solve a partial differential
More informationPart IV: Numerical schemes for the phase-filed model
Part IV: Numerical schemes for the phase-filed model Jie Shen Department of Mathematics Purdue University IMS, Singapore July 29-3, 29 The complete set of governing equations Find u, p, (φ, ξ) such that
More informationITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS
ITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS LONG CHEN In this chapter we discuss iterative methods for solving the finite element discretization of semi-linear elliptic equations of the form: find
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik On the stable discretization of strongly anisotropic phase field models with applications to crystal growth John W. Barrett, Harald Garcke and Robert Nürnberg Preprint
More informationTHE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS
THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS Alain Miranville Université de Poitiers, France Collaborators : L. Cherfils, G. Gilardi, G.R. Goldstein, G. Schimperna,
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 informationA PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM
A PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM Hyun Geun LEE 1, Jeong-Whan CHOI 1 and Junseok KIM 1 1) Department of Mathematics, Korea University, Seoul
More informationVARIATIONAL AND NON-VARIATIONAL MULTIGRID ALGORITHMS FOR THE LAPLACE-BELTRAMI OPERATOR.
VARIATIONAL AND NON-VARIATIONAL MULTIGRID ALGORITHMS FOR THE LAPLACE-BELTRAMI OPERATOR. ANDREA BONITO AND JOSEPH E. PASCIAK Abstract. We design and analyze variational and non-variational multigrid algorithms
More informationNumerical explorations of a forward-backward diffusion equation
Numerical explorations of a forward-backward diffusion equation Newton Institute KIT Programme Pauline Lafitte 1 C. Mascia 2 1 SIMPAF - INRIA & U. Lille 1, France 2 Univ. La Sapienza, Roma, Italy September
More informationEntropy-dissipation methods I: Fokker-Planck equations
1 Entropy-dissipation methods I: Fokker-Planck equations Ansgar Jüngel Vienna University of Technology, Austria www.jungel.at.vu Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic
More informationNumerical Modeling of Methane Hydrate Evolution
Numerical Modeling of Methane Hydrate Evolution Nathan L. Gibson Joint work with F. P. Medina, M. Peszynska, R. E. Showalter Department of Mathematics SIAM Annual Meeting 2013 Friday, July 12 This work
More informationarxiv: v1 [cs.na] 13 Jan 2016
An Optimal Block Diagonal Preconditioner for Heterogeneous Saddle Point Problems in Phase Separation Pawan Kumar 1 arxiv:1601.03230v1 [cs.na] 13 Jan 2016 1 Arnimallee 6 1 Department of Mathematics and
More informationEfficient smoothers for all-at-once multigrid methods for Poisson and Stokes control problems
Efficient smoothers for all-at-once multigrid methods for Poisson and Stoes control problems Stefan Taacs stefan.taacs@numa.uni-linz.ac.at, WWW home page: http://www.numa.uni-linz.ac.at/~stefant/j3362/
More informationA MULTILEVEL SUCCESSIVE ITERATION METHOD FOR NONLINEAR ELLIPTIC PROBLEMS
MATHEMATICS OF COMPUTATION Volume 73, Number 246, Pages 525 539 S 0025-5718(03)01566-7 Article electronically published on July 14, 2003 A MULTILEVEL SUCCESSIVE ITERATION METHOD FOR NONLINEAR ELLIPTIC
More informationEvolution Under Constraints: Fate of Methane in Subsurface
Evolution Under Constraints: Fate of Methane in Subsurface M. Peszyńska 1 Department of Mathematics, Oregon State University SIAM PDE. Nov.2011 1 Research supported by DOE 98089 Modeling, Analysis, and
More informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1. Introduction: Direct solvers versus iterative solvers In many applications we have to solve a linear system Ax = b with A R n n and b R n given. If n is large the
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationDiscrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction Problem Jörg-M. Sautter Mathematisches Institut, Universität Düsseldorf, Germany, sautter@am.uni-duesseldorf.de
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 informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationAdaptive Finite Element Methods Lecture Notes Winter Term 2017/18. R. Verfürth. Fakultät für Mathematik, Ruhr-Universität Bochum
Adaptive Finite Element Methods Lecture Notes Winter Term 2017/18 R. Verfürth Fakultät für Mathematik, Ruhr-Universität Bochum Contents Chapter I. Introduction 7 I.1. Motivation 7 I.2. Sobolev and finite
More information1. Fast Iterative Solvers of SLE
1. Fast Iterative Solvers of crucial drawback of solvers discussed so far: they become slower if we discretize more accurate! now: look for possible remedies relaxation: explicit application of the multigrid
More informationA Sobolev trust-region method for numerical solution of the Ginz
A Sobolev trust-region method for numerical solution of the Ginzburg-Landau equations Robert J. Renka Parimah Kazemi Department of Computer Science & Engineering University of North Texas June 6, 2012
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationGeometric Multigrid Methods
Geometric Multigrid Methods Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University IMA Tutorial: Fast Solution Techniques November 28, 2010 Ideas
More informationKasetsart University Workshop. Multigrid methods: An introduction
Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available
More informationNumerical Simulations on Two Nonlinear Biharmonic Evolution Equations
Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations Ming-Jun Lai, Chun Liu, and Paul Wenston Abstract We numerically simulate the following two nonlinear evolution equations with a fourth
More informationSmall energy regularity for a fractional Ginzburg-Landau system
Small energy regularity for a fractional Ginzburg-Landau system Yannick Sire University Aix-Marseille Work in progress with Vincent Millot (Univ. Paris 7) The fractional Ginzburg-Landau system We are interest
More informationBoundary Value Problems and Iterative Methods for Linear Systems
Boundary Value Problems and Iterative Methods for Linear Systems 1. Equilibrium Problems 1.1. Abstract setting We want to find a displacement u V. Here V is a complete vector space with a norm v V. In
More informationA Line search Multigrid Method for Large-Scale Nonlinear Optimization
A Line search Multigrid Method for Large-Scale Nonlinear Optimization Zaiwen Wen Donald Goldfarb Department of Industrial Engineering and Operations Research Columbia University 2008 Siam Conference on
More informationNUCLEATION AND SPINODAL DECOMPOSITION IN TERNARY-COMPONENT ALLOYS
NUCLEATION AND SPINODAL DECOMPOSITION IN TERNARY-COMPONENT ALLOYS WILL HARDESTY AND COLLEEN ACKERMANN Date: July 8, 009. Dr. Thomas Wanner and Dr. Evelyn Sander. 0 Abstract. put abstract here NUCLEATION
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationarxiv: v2 [math.na] 17 Jun 2010
Numerische Mathematik manuscript No. (will be inserted by the editor) Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids Long Chen 1, Michael Holst 2, Jinchao
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 informationGoal-oriented error control of the iterative solution of finite element equations
J. Numer. Math., Vol. 0, No. 0, pp. 1 31 (2009) DOI 10.1515/ JNUM.2009.000 c de Gruyter 2009 Prepared using jnm.sty [Version: 20.02.2007 v1.3] Goal-oriented error control of the iterative solution of finite
More informationGoal-oriented error control of the iterative solution of finite element equations
J. Numer. Math., Vol. 17, No. 2, pp. 143 172 (2009) DOI 10.1515/ JNUM.2009.009 c de Gruyter 2009 Goal-oriented error control of the iterative solution of finite element equations D. MEIDNER, R. RANNACHER,
More informationCONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS
CONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS Igor V. Konnov Department of Applied Mathematics, Kazan University Kazan 420008, Russia Preprint, March 2002 ISBN 951-42-6687-0 AMS classification:
More informationA C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. Marco Verani
A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes Marco Verani MOX, Department of Mathematics, Politecnico di Milano Joint work with: P. F. Antonietti (MOX Politecnico di
More informationChapter 7 Iterative Techniques in Matrix Algebra
Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition
More informationMULTIMATERIAL TOPOLOGY OPTIMIZATION BY VOLUME CONSTRAINED ALLEN-CAHN SYSTEM AND REGULARIZED PROJECTED STEEPEST DESCENT METHOD
MULTIMATERIAL TOPOLOGY OPTIMIZATION BY VOLUME CONSTRAINED ALLEN-CAHN SYSTEM AND REGULARIZED PROJECTED STEEPEST DESCENT METHOD R. TAVAKOLI Abstract. A new computational algorithm is introduced in the present
More informationWeak solutions for the Cahn-Hilliard equation with degenerate mobility
Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor) Shibin Dai Qiang Du Weak solutions for the Cahn-Hilliard equation with degenerate mobility Abstract In this paper,
More informationNumerical Solution of Nonsmooth Problems and Application to Damage Evolution and Optimal Insulation
Numerical Solution of Nonsmooth Problems and Application to Damage Evolution and Optimal Insulation Sören Bartels University of Freiburg, Germany Joint work with G. Buttazzo (U Pisa), L. Diening (U Bielefeld),
More informationarxiv: v1 [math.na] 11 Jul 2011
Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients arxiv:07.260v [math.na] Jul 20 Blanca Ayuso De Dios, Michael Holst 2, Yunrong Zhu 2, and Ludmil Zikatanov
More informationA generalized MBO diffusion generated motion for constrained harmonic maps
A generalized MBO diffusion generated motion for constrained harmonic maps Dong Wang Department of Mathematics, University of Utah Joint work with Braxton Osting (U. Utah) Workshop on Modeling and Simulation
More informationApproximation of fluid-structure interaction problems with Lagrange multiplier
Approximation of fluid-structure interaction problems with Lagrange multiplier Daniele Boffi Dipartimento di Matematica F. Casorati, Università di Pavia http://www-dimat.unipv.it/boffi May 30, 2016 Outline
More informationApplied/Numerical Analysis Qualifying Exam
Applied/Numerical Analysis Qualifying Exam August 9, 212 Cover Sheet Applied Analysis Part Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless,
More informationNonlinear Multigrid and Domain Decomposition Methods
Nonlinear Multigrid and Domain Decomposition Methods Rolf Krause Institute of Computational Science Universit`a della Svizzera italiana 1 Stepping Stone Symposium, Geneva, 2017 Non-linear problems Non-linear
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 informationMultigrid and Domain Decomposition Methods for Electrostatics Problems
Multigrid and Domain Decomposition Methods for Electrostatics Problems Michael Holst and Faisal Saied Abstract. We consider multigrid and domain decomposition methods for the numerical solution of electrostatics
More informationUNCONDITIONALLY ENERGY STABLE SCHEME FOR CAHN-MORRAL EQUATION 3
UNCONDITIONALLY ENERGY STABLE TIME STEPPING SCHEME FOR CAHN-MORRAL EQUATION: APPLICATION TO MULTI-COMPONENT SPINODAL DECOMPOSITION AND OPTIMAL SPACE TILING R. TAVAKOLI Abstract. An unconditionally energy
More informationA SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS
A SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS MICHAEL HOLST AND FAISAL SAIED Abstract. We consider multigrid and domain decomposition methods for the numerical
More informationAn additive average Schwarz method for the plate bending problem
J. Numer. Math., Vol. 10, No. 2, pp. 109 125 (2002) c VSP 2002 Prepared using jnm.sty [Version: 02.02.2002 v1.2] An additive average Schwarz method for the plate bending problem X. Feng and T. Rahman Abstract
More informationA STOKES INTERFACE PROBLEM: STABILITY, FINITE ELEMENT ANALYSIS AND A ROBUST SOLVER
European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.) O. Nevanlinna and R. Rannacher (assoc. eds.) Jyväskylä,
More informationSpace-time sparse discretization of linear parabolic equations
Space-time sparse discretization of linear parabolic equations Roman Andreev August 2, 200 Seminar for Applied Mathematics, ETH Zürich, Switzerland Support by SNF Grant No. PDFMP2-27034/ Part of PhD thesis
More informationTitle of your thesis here
Title of your thesis here Your name here A Thesis presented for the degree of Doctor of Philosophy Your Research Group Here Department of Mathematical Sciences University of Durham England Month and Year
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 informationMultigrid Methods and their application in CFD
Multigrid Methods and their application in CFD Michael Wurst TU München 16.06.2009 1 Multigrid Methods Definition Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential
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 informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationMulti-Factor Finite Differences
February 17, 2017 Aims and outline Finite differences for more than one direction The θ-method, explicit, implicit, Crank-Nicolson Iterative solution of discretised equations Alternating directions implicit
More informationCoupled second order singular perturbations for phase transitions
Coupled second order singular perturbations for phase transitions CMU 06/09/11 Ana Cristina Barroso, Margarida Baía, Milena Chermisi, JM Introduction Let Ω R d with Lipschitz boundary ( container ) and
More informationNumerical Oscillations and how to avoid them
Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A.
More informationStability Analysis of Stationary Solutions for the Cahn Hilliard Equation
Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation Peter Howard, Texas A&M University University of Louisville, Oct. 19, 2007 References d = 1: Commun. Math. Phys. 269 (2007) 765
More informationA Space-Time Multigrid Solver Methodology for the Optimal Control of Time-Dependent Fluid Flow
A Space-Time Multigrid Solver Methodology for the Optimal Control of Time-Dependent Fluid Flow Michael Köster, Michael Hinze, Stefan Turek Michael Köster Institute for Applied Mathematics TU Dortmund Trier,
More informationThe method of lines (MOL) for the diffusion equation
Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just
More informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationMulti-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities
Multi-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities R. Kornhuber and C. Schwab and M. Wolf Research Report No. 2013-12 April 2013 Seminar für Angewandte Mathematik
More informationOn Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability
On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability Ming-Jun Lai, Chun Liu, and Paul Wenston Abstract We study the following two nonlinear evolution equations with a fourth
More informationVariational and Topological methods : Theory, Applications, Numerical Simulations, and Open Problems 6-9 June 2012, Northern Arizona University
Variational and Topological methods : Theory, Applications, Numerical Simulations, and Open Problems 6-9 June 22, Northern Arizona University Some methods using monotonicity for solving quasilinear parabolic
More informationDiscontinuous Galerkin methods for fractional diffusion problems
Discontinuous Galerkin methods for fractional diffusion problems Bill McLean Kassem Mustapha School of Maths and Stats, University of NSW KFUPM, Dhahran Leipzig, 7 October, 2010 Outline Sub-diffusion Equation
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 informationPriority Programme The Combination of POD Model Reduction with Adaptive Finite Element Methods in the Context of Phase Field Models
Priority Programme 1962 The Combination of POD Model Reduction with Adaptive Finite Element Methods in the Context of Phase Field Models Carmen Gräßle, Michael Hinze Non-smooth and Complementarity-based
More informationSimulating Solid Tumor Growth Using Multigrid Algorithms
Simulating Solid Tumor Growth Using Multigrid Algorithms Asia Wyatt Applied Mathematics, Statistics, and Scientific Computation Program Advisor: Doron Levy Department of Mathematics/CSCAMM Abstract In
More informationA New Approach for the Numerical Solution of Diffusion Equations with Variable and Degenerate Mobility
A New Approach for the Numerical Solution of Diffusion Equations with Variable and Degenerate Mobility Hector D. Ceniceros Department of Mathematics, University of California Santa Barbara, CA 93106 Carlos
More informationNumerical Analysis of Differential Equations Numerical Solution of Parabolic Equations
Numerical Analysis of Differential Equations 215 6 Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012 Numerical Analysis of Differential
More informationPDEs in Image Processing, Tutorials
PDEs in Image Processing, Tutorials Markus Grasmair Vienna, Winter Term 2010 2011 Direct Methods Let X be a topological space and R: X R {+ } some functional. following definitions: The mapping R is lower
More informationSolving Distributed Optimal Control Problems for the Unsteady Burgers Equation in COMSOL Multiphysics
Excerpt from the Proceedings of the COMSOL Conference 2009 Milan Solving Distributed Optimal Control Problems for the Unsteady Burgers Equation in COMSOL Multiphysics Fikriye Yılmaz 1, Bülent Karasözen
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 4: Iterative Methods PD
More informationNumerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation
More informationParameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition
Parameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition P. A. Farrell 1,A.F.Hegarty 2, J. J. H. Miller 3, E. O Riordan 4,andG.I. Shishkin 5 1
More informationAn Iterative Substructuring Method for Mortar Nonconforming Discretization of a Fourth-Order Elliptic Problem in two dimensions
An Iterative Substructuring Method for Mortar Nonconforming Discretization of a Fourth-Order Elliptic Problem in two dimensions Leszek Marcinkowski Department of Mathematics, Warsaw University, Banacha
More informationSubdiffusion in a nonconvex polygon
Subdiffusion in a nonconvex polygon Kim Ngan Le and William McLean The University of New South Wales Bishnu Lamichhane University of Newcastle Monash Workshop on Numerical PDEs, February 2016 Outline Time-fractional
More informationNEWTON-GMRES PRECONDITIONING FOR DISCONTINUOUS GALERKIN DISCRETIZATIONS OF THE NAVIER-STOKES EQUATIONS
NEWTON-GMRES PRECONDITIONING FOR DISCONTINUOUS GALERKIN DISCRETIZATIONS OF THE NAVIER-STOKES EQUATIONS P.-O. PERSSON AND J. PERAIRE Abstract. We study preconditioners for the iterative solution of the
More informationA domain decomposition algorithm for contact problems with Coulomb s friction
A domain decomposition algorithm for contact problems with Coulomb s friction J. Haslinger 1,R.Kučera 2, and T. Sassi 1 1 Introduction Contact problems of elasticity are used in many fields of science
More informationAllen Cahn Equation in Two Spatial Dimension
Allen Cahn Equation in Two Spatial Dimension Yoichiro Mori April 25, 216 Consider the Allen Cahn equation in two spatial dimension: ɛ u = ɛ2 u + fu) 1) where ɛ > is a small parameter and fu) is of cubic
More informationNumerical Solution of PDEs: Bounds for Functional Outputs and Certificates
Numerical Solution of PDEs: Bounds for Functional Outputs and Certificates J. Peraire Massachusetts Institute of Technology, USA ASC Workshop, 22-23 August 2005, Stanford University People MIT: A.T. Patera,
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Optimal control of Allen-Cahn systems Luise Blank, M. Hassan Farshbaf-Shaker, Claudia Hecht Josef Michl and Christoph Rupprecht Preprint Nr. 21/2013 Optimal control of
More informationMultigrid finite element methods on semi-structured triangular grids
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, -5 septiembre 009 (pp. 8) Multigrid finite element methods on semi-structured triangular grids F.J.
More informationA posteriori error estimation for elliptic problems
A posteriori error estimation for elliptic problems Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in
More information