IFE for Stokes interface problem
|
|
- Brittney Washington
- 5 years ago
- Views:
Transcription
1 IFE for Stokes interface problem Nabil Chaabane Slimane Adjerid, Tao Lin Virginia Tech SIAM chapter February 4, 24 Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 / 2
2 Problem statement Consider the Stokes problem where the Stress tensor is S(u, p) = f, in Ω, (a) u =, in Ω, (b) u = g, on Ω, (c) S lj (u, p) = ν(d l u j + D j u l ) pδ lj, l, j =, 2. Ω is occupied by two fluids ν + and ν separated by an interface Γ. The jump conditions are [S(u, p) n] Γ =, (2) [u] Γ =. (3) In this work, we assume that the pressure is continuous. Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 2 / 2
3 Ω Γ Ω + Ω Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 3 / 2
4 Ω Γ Ω + Ω Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 4 / 2
5 IFE basis functions We approximate the interface with a straight line. We map the physical element to the reference element using appropriate mapping. Then, we construct a piecewise vector-valued function ˆΦ s.t: { ˆΦ ˆΦ + if (ˆx, ŷ) ˆT + = ˆΦ if (ˆx, ŷ) ˆT, Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 5 / 2
6 IFE basis functions ˆΦ (ˆx, ŷ) = ( a + b ˆx + c ŷ + d ˆxŷ a 2 + b 2 ˆx + c 2 ŷ + d 2 ˆxŷ ˆΦ + (ˆx, ŷ) = ( a + + b + ˆx + c+ ŷ + d + ˆxŷ a b+ 2 ˆx + c+ 2 ŷ + d + 2 ˆxŷ The IFE basis functions satisfy the following jump constraints ), (4a) ), (4b) ˆΦ (Ê) = ˆΦ + (Ê), ˆΦ j (Â i ) = δ ij, i, j =, 2,..., 8, (5) ˆΦ ( ˆD) = ˆΦ + ( ˆD), 2 ˆΦ ˆx ŷ = 2 ˆΦ + ˆx ŷ, (6) [S( ˆΦ, p) n ]ds =. (7) ˆDÊ ˆDÊ The jump conditions yield 6 constraints that are used to solve for the unknowns. Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 6 / 2
7 Properties of the bilinear IFE space On every interface element T the IFE basis: are well defined and are unique. are continuous. form a partition of unity i.e 4 Φ i (x, y) = i= ( ), 8 Φ i (x, y) = i=4 ( ), (x, y) t T. Furthermore, If the viscosity parameters have no discontinuity i.e ν + = ν, then Φ i becomes the standard vector-valued bilinear nodal basis Ψ i. If min{ T +, T } shrinks to, then Φ i becomes the standard vector bilinear nodal basis Ψ i. The interpolation error converges with optimal rate. Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 7 / 2
8 IFE basis functions Figure: First and second components of the IFE basis functions with ν =, ν + = 5.3 on the left. The first and second components of the standard nodal finite element basis on the right. Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 8 / 2
9 Interpolation error h u I h u,ω Order u 2 I h u 2,Ω Order.6844e-.6844e e e e e e e e e e e-3.37 Table: L 2 error of the interpolation error with ν = and ν + = 5.3 Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 9 / 2
10 Interpolation error h u (I h u),ω Order u 2 (I h u) 2,Ω Order e e e e e e e e e e Table: L 2 error of the interpolation error with ν = and ν + = 5.3 h u (I h u),ω Order u 2 (I h u) 2,Ω Order e e e e e e e e e e Table: H error of the interpolation error with ν = and ν + = 5.3 Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 / 2
11 IFE spaces using Q 2 iso Q /Q elements Consider an arbitrary uniform cartesian mesh T h and its refinement T h/2 and define the following on T h/2 { span{φ i (x, y), i =, 2,..., 8}, T is an interface element S h/2 (T ) = span{ψ i (x, y), i =, 2,..., 8}, otherwise, and define S h/2 (Ω) = T Th/2 S h/2 (T ), and S h/2, (Ω) = {v S h/2 (Ω) : v e =, e Eh/2 B \Ei h/2 }, where Eh/2 i is the set of interface edges. On the mesh T h, define M h (Ω) = {q C L 2 : q T Q }. Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 / 2
12 IFE formulation The finite element formulation is: Find (u h, p h ) S h (Ω) M h (Ω) s.t 2 S(u h, p h ) : v h dx = f v h dx, v h S h,(ω) (8a) 2 Ω Ω u h q h dx =, q h M h (Ω), (8b) Ω Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 2 / 2
13 Numerical results The domain Ω is the square [, ] 2, the interface is a half-circle with radius r =.3, and center x =.4327, y =.329 that separates Ω into two regions Ω + = {(x, y) t : (x x ) 2 + (y y ) 2 > r 2 } and Ω = {(x, y) t : (x x ) 2 + (y y ) 2 < r 2 }. The function u is defined as { u + u + (x, y) = = ((x x ) 2 + (y y ) 2 r 2 )(y y ), u 2 + = ((x x ) 2 + (y y ) 2 r 2, (9a) )(x x ), u = ν+ ν u+. (9b) Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 3 / 2
14 Numerical results h u u,h,ω u 2 u 2,h,Ω p p h,ω e e e e e e e e e e e e e e e e e e-2 Table: L 2 errors with ν = and ν + = 5.3 for the non-penalized method. u u,h,ω Ch.587, u 2 u 2,h,Ω Ch.6484, p p h,ω Ch Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 4 / 2
15 Figure: Standard Lagrange finite element basis Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 5 / 2
16 Figure: Immersed finite element basis Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 6 / 2
17 IFE formulation The finite element formulation is: Find (u h, p h ) S h (Ω) M h (Ω) s.t 2 S(u h, p h ) : v h dx Ω e Eh i + e E i h \EB h e e E i h \EB h [u h ] [v h ]ds = Ω Ω e {(S(u h, p h ) n)} [v h ]ds e {(S(v h, ) n)} [u h ]ds f v h dx, (a) v h S h,(ω) (b) 2 u h q h dx =, q h M h (Ω), (c) Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 7 / 2
18 Numerical results The domain Ω is the square [, ] 2, the interface is a half-circle with radius r =.3, and center x =.4327, y =.329 that separates Ω into two regions Ω + = {(x, y) t : (x x ) 2 + (y y ) 2 > r 2 } and Ω = {(x, y) t : (x x ) 2 + (y y ) 2 < r 2 }. The function u is defined as { u + u + (x, y) = = ((x x ) 2 + (y y ) 2 r 2 )(y y ), u 2 + = ((x x ) 2 + (y y ) 2 r 2, (a) )(x x ), u = ν+ ν u+. (b) Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 8 / 2
19 Numerical results h u u,h,ω Order u 2 u 2,h,Ω Order.592e e e e e e e e e e e e-6.99 Table: L 2 error and order of convergence with ν = and ν + = 5.3 Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 9 / 2
20 Numerical results h p p h,ω Order 5.363e e e e e e-4.45 Table: L 2 error and order of convergence with ν = and ν + = 5.3 Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 2 / 2
21 THANK YOU! Nabil Chaabane (VT) IFE for Stokes interface problem February 4, 24 2 / 2
HIGH DEGREE IMMERSED FINITE ELEMENT SPACES BY A LEAST SQUARES METHOD
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 14, Number 4-5, Pages 604 626 c 2017 Institute for Scientific Computing and Information HIGH DEGREE IMMERSED FINITE ELEMENT SPACES BY A LEAST
More informationBilinear Immersed Finite Elements For Interface Problems
Bilinear Immersed Finite Elements For Interface Problems Xiaoming He Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Nonconformity and the Consistency Error First Strang Lemma Abstract Error Estimate
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 p-th Degree Immersed Finite Element for Boundary Value Problems with Discontinuous Coefficients
A p-th Degree Immersed Finite Element for Boundary Value Problems with Discontinuous Coefficients Slimane Adjerid and Tao Lin Department of Mathematics Virginia Polytechnic Institute and State University
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationPartially Penalized Immersed Finite Element Methods for Parabolic Interface Problems
Partially Penalized Immersed Finite Element Methods for Parabolic Interface Problems Tao Lin, Qing Yang and Xu Zhang Abstract We present partially penalized immersed finite element methods for solving
More information1 Discretizing BVP with Finite Element Methods.
1 Discretizing BVP with Finite Element Methods In this section, we will discuss a process for solving boundary value problems numerically, the Finite Element Method (FEM) We note that such method is a
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationAn interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes
An interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes Vincent Heuveline Friedhelm Schieweck Abstract We propose a Scott-Zhang type interpolation
More informationChapter 3 Conforming Finite Element Methods 3.1 Foundations Ritz-Galerkin Method
Chapter 3 Conforming Finite Element Methods 3.1 Foundations 3.1.1 Ritz-Galerkin Method Let V be a Hilbert space, a(, ) : V V lr a bounded, V-elliptic bilinear form and l : V lr a bounded linear functional.
More informationAdaptive C1 Macroelements for Fourth Order and Divergence-Free Problems
Adaptive C1 Macroelements for Fourth Order and Divergence-Free Problems Roy Stogner Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin March
More informationA P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS
A P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS SHANGYOU ZHANG DEDICATED TO PROFESSOR PETER MONK ON THE OCCASION OF HIS 6TH BIRTHDAY Abstract. On triangular grids, the continuous
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 informationAn Extended Finite Element Method for a Two-Phase Stokes problem
XFEM project An Extended Finite Element Method for a Two-Phase Stokes problem P. Lederer, C. Pfeiler, C. Wintersteiger Advisor: Dr. C. Lehrenfeld August 5, 2015 Contents 1 Problem description 2 1.1 Physics.........................................
More informationUniversity of Illinois at Chicago Department of Physics
University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.
More informationConstrained H 1 -interpolation on quadrilateral and hexahedral meshes with hanging nodes
Constrained H 1 -interpolation on quadrilateral and hexahedral meshes with hanging nodes Vincent Heuveline Friedhelm Schieweck Abstract We propose a Scott-Zhang type finite element interpolation operator
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 informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationA POSTERIORI ERROR ESTIMATION FOR NON-CONFORMING QUADRILATERAL FINITE ELEMENTS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 2, Number, Pages 8 c 2005 Institute for Scientific Computing and Information A POSTERIORI ERROR ESTIMATION FOR NON-CONFORMING QUADRILATERAL
More informationFinite Element Methods for Maxwell Equations
CHAPTER 8 Finite Element Methods for Maxwell Equations The Maxwell equations comprise four first-order partial differential equations linking the fundamental electromagnetic quantities, the electric field
More informationOverlapping Schwarz preconditioners for Fekete spectral elements
Overlapping Schwarz preconditioners for Fekete spectral elements R. Pasquetti 1, L. F. Pavarino 2, F. Rapetti 1, and E. Zampieri 2 1 Laboratoire J.-A. Dieudonné, CNRS & Université de Nice et Sophia-Antipolis,
More informationANALYSIS AND NUMERICAL METHODS FOR SOME CRACK PROBLEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 2, Number 2-3, Pages 155 166 c 2011 Institute for Scientific Computing and Information ANALYSIS AND NUMERICAL METHODS FOR SOME
More informationLeast-Squares Spectral Collocation with the Overlapping Schwarz Method for the Incompressible Navier Stokes Equations
Least-Squares Spectral Collocation with the Overlapping Schwarz Method for the Incompressible Navier Stokes Equations by Wilhelm Heinrichs Universität Duisburg Essen, Ingenieurmathematik Universitätsstr.
More informationWell-balanced DG scheme for Euler equations with gravity
Well-balanced DG scheme for Euler equations with gravity Praveen Chandrashekar praveen@tifrbng.res.in Center for Applicable Mathematics Tata Institute of Fundamental Research Bangalore 560065 Dept. of
More informationIsogeometric Analysis:
Isogeometric Analysis: some approximation estimates for NURBS L. Beirao da Veiga, A. Buffa, Judith Rivas, G. Sangalli Euskadi-Kyushu 2011 Workshop on Applied Mathematics BCAM, March t0th, 2011 Outline
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 informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationExplicit Jump Immersed Interface Method: Documentation for 2D Poisson Code
Eplicit Jump Immersed Interface Method: Documentation for 2D Poisson Code V. Rutka A. Wiegmann November 25, 2005 Abstract The Eplicit Jump Immersed Interface method is a powerful tool to solve elliptic
More informationICES REPORT A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams
ICS RPORT 15-17 July 2015 A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams by Omar Al-Hinai, Mary F. Wheeler, Ivan Yotov The Institute for Computational ngineering
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 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 informationStochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows
Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows Ivan otov Department of Mathematics, University of Pittsburgh
More informationPAijpam.eu NEW H 1 (Ω) CONFORMING FINITE ELEMENTS ON HEXAHEDRA
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 609-617 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v109i3.10
More informationHEXAHEDRAL H(DIV) AND H(CURL) FINITE ELEMENTS
HEXAHEDRAL H(DIV) AND H(CURL) FINITE ELEMENTS RICHARD S. FAL, PAOLO GATTO, AND PETER MON Abstract. We study the approximation properties of some finite element subspaces of H(div; Ω) and H(curl ; Ω) defined
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 informationHybridized Discontinuous Galerkin Methods
Hybridized Discontinuous Galerkin Methods Theory and Christian Waluga Joint work with Herbert Egger (Uni Graz) 1st DUNE User Meeting, Stuttgart Christian Waluga (AICES) HDG Methods October 6-8, 2010 1
More informationA Balancing Algorithm for Mortar Methods
A Balancing Algorithm for Mortar Methods Dan Stefanica Baruch College, City University of New York, NY 11, USA Dan Stefanica@baruch.cuny.edu Summary. The balancing methods are hybrid nonoverlapping Schwarz
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 informationChapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationWeak Galerkin Finite Element Scheme and Its Applications
Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015 Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer
More informationSolving an elasto-plastic model using DOLFIN
Solving an elasto-plastic model using DOLFIN TTI 2005 Johan Jansson johanjan@math.chalmers.se Chalmers University of Technology Solving an elasto-plastic model using DOLFIN p. Overview Motivation Previous
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationWell-balanced DG scheme for Euler equations with gravity
Well-balanced DG scheme for Euler equations with gravity Praveen Chandrashekar praveen@tifrbng.res.in Center for Applicable Mathematics Tata Institute of Fundamental Research Bangalore 560065 Higher Order
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 informationOn the hydrodynamic diffusion of rigid particles
On the hydrodynamic diffusion of rigid particles O. Gonzalez Introduction Basic problem. Characterize how the diffusion and sedimentation properties of particles depend on their shape. Diffusion: Sedimentation:
More informationYongdeok Kim and Seki Kim
J. Korean Math. Soc. 39 (00), No. 3, pp. 363 376 STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM Yongdeok Kim and Seki Kim Abstract. Stability result is obtained for
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Variational Problems of the Dirichlet BVP of the Poisson Equation 1 For the homogeneous
More informationUniversity of Groningen
University of Groningen Nature-inspired microfluidic propulsion using magnetic actuation Khaderi, S. N.; Baltussen, M. G. H. M.; Anderson, P. D.; Ioan, D.; den Toonder, J.M.J.; Onck, Patrick Published
More informationA multipoint flux mixed finite element method on hexahedra
A multipoint flux mixed finite element method on hexahedra Ross Ingram Mary F. Wheeler Ivan Yotov Abstract We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationSpace-time XFEM for two-phase mass transport
Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague,
More informationFast evaluation of boundary integral operators arising from an eddy current problem. MPI MIS Leipzig
1 Fast evaluation of boundary integral operators arising from an eddy current problem Steffen Börm MPI MIS Leipzig Jörg Ostrowski ETH Zürich Induction heating 2 Induction heating: Time harmonic eddy current
More informationABHELSINKI UNIVERSITY OF TECHNOLOGY
ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for the Morley plate element Jarkko Niiranen Department of Structural
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationNew class of finite element methods: weak Galerkin methods
New class of finite element methods: weak Galerkin methods Xiu Ye University of Arkansas at Little Rock Second order elliptic equation Consider second order elliptic problem: a u = f, in Ω (1) u = 0, on
More informationICES REPORT Direct Serendipity Finite Elements on Convex Quadrilaterals
ICES REPORT 17-8 October 017 Direct Serendipity Finite Elements on Convex Quadrilaterals by Todd Arbogast and Zhen Tao The Institute for Computational Engineering and Sciences The University of Texas at
More information- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20 - Marine Hydrodynamics, Spring 2005 Lecture 4 2.20 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationCONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM
CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM Summary of integral theorems Material time derivative Reynolds transport theorem Principle of conservation of mass Principle of balance of linear momentum
More informationThe Plane Stress Problem
The Plane Stress Problem Martin Kronbichler Applied Scientific Computing (Tillämpad beräkningsvetenskap) February 2, 2010 Martin Kronbichler (TDB) The Plane Stress Problem February 2, 2010 1 / 24 Outline
More informationDG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement
16th International Conference on Finite Elements in Flow Problems DG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement Institute of Aerodynamics and Flow Technology
More informationAnalysis of a high order trace finite element method for PDEs on level set surfaces
N O V E M B E R 2 0 1 6 P R E P R I N T 4 5 7 Analysis of a high order trace finite element method for PDEs on level set surfaces Jörg Grande *, Christoph Lehrenfeld and Arnold Reusken * Institut für Geometrie
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 informationThe Discontinuous Galerkin Finite Element Method
The Discontinuous Galerkin Finite Element Method Michael A. Saum msaum@math.utk.edu Department of Mathematics University of Tennessee, Knoxville The Discontinuous Galerkin Finite Element Method p.1/41
More informationA Two-grid Method for Coupled Free Flow with Porous Media Flow
A Two-grid Method for Coupled Free Flow with Porous Media Flow Prince Chidyagwai a and Béatrice Rivière a, a Department of Computational and Applied Mathematics, Rice University, 600 Main Street, Houston,
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationSome remarks on grad-div stabilization of incompressible flow simulations
Some remarks on grad-div stabilization of incompressible flow simulations Gert Lube Institute for Numerical and Applied Mathematics Georg-August-University Göttingen M. Stynes Workshop Numerical Analysis
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 informationHp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme
Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme Aravind Balan, Michael Woopen and Georg May AICES Graduate School, RWTH Aachen University, Germany 22nd AIAA Computational
More information(2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)?
Part I: Quantum Mechanics: Principles & Models 1. General Concepts: (2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)? (4 pts) b. How does
More informationPREPRINT 2010:23. A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO
PREPRINT 2010:23 A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationA brief introduction to finite element methods
CHAPTER A brief introduction to finite element methods 1. Two-point boundary value problem and the variational formulation 1.1. The model problem. Consider the two-point boundary value problem: Given a
More informationC e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a
C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and Simulation Modelling, Analysis and Simulation Bilinear forms for the recovery-based discontinuous Galerkin method
More informationAn explicit nite element method for convection-dominated compressible viscous Stokes system with inow boundary
Journal of Computational and Applied Mathematics 156 (2003) 319 343 www.elsevier.com/locate/cam An explicit nite element method for convection-dominated compressible viscous Stokes system with inow boundary
More informationTwo Nonconforming Quadrilateral Elements for the Reissner-Mindlin Plate
Two Nonconforming Quadrilateral Elements for the Reissner-Mindlin Plate Pingbing Ming and Zhong-ci Shi Institute of Computational Mathematics & Scientific/Engineering Computing, AMSS, Chinese Academy of
More informationThe Kolmogorov Law of turbulence
What can rigorously be proved? IRMAR, UMR CNRS 6625. Labex CHL. University of RENNES 1, FRANCE Introduction Aim: Mathematical framework for the Kolomogorov laws. Table of contents 1 Incompressible Navier-Stokes
More informationMultigrid Methods for Maxwell s Equations
Multigrid Methods for Maxwell s Equations Jintao Cui Institute for Mathematics and Its Applications University of Minnesota Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl
More information2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationA note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations
A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations Bernardo Cockburn Guido anschat Dominik Schötzau June 1, 2007 Journal of Scientific Computing, Vol. 31, 2007, pp.
More informationZero Energy Modes in One Dimension: An Introduction to Hourglass Modes
Zero Energy Modes in One Dimension: An Introduction to Hourglass Modes David J. Benson March 9, 2003 Reduced integration does a lot of good things for an element: it reduces the computational cost, it
More informationFinite volume method for two-phase flows using level set formulation
Finite volume method for two-phase flows using level set formulation Peter Frolkovič,a,1, Dmitry Logashenko b, Christian Wehner c, Gabriel Wittum c a Department of Mathematics and Descriptive Geometry,
More information7 Curvilinear coordinates
7 Curvilinear coordinates Read: Boas sec. 5.4, 0.8, 0.9. 7. Review of spherical and cylindrical coords. First I ll review spherical and cylindrical coordinate systems so you can have them in mind when
More informationAND BARBARA I. WOHLMUTH
A QUASI-DUAL LAGRANGE MULTIPLIER SPACE FOR SERENDIPITY MORTAR FINITE ELEMENTS IN 3D BISHNU P. LAMICHHANE AND BARBARA I. WOHLMUTH Abstract. Domain decomposition techniques provide a flexible tool for the
More informationC 0 P 2 P 0 Stokes finite element pair on sub-hexahedron tetrahedral grids
Calcolo 2017 54:1403 1417 https://doi.org/10.1007/s1002-017-0235-2 C 0 P 2 P 0 Stokes finite element pair on sub-hexahedron tetrahedral grids Shangyou Zhang 1 Shuo Zhang 2 Received: 22 March 2017 / Accepted:
More informationPROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO
PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO Overview: 1. Motivation 1.1. Evolutionary problem of crack propagation 1.2. Stationary problem of crack equilibrium 1.3. Interaction (contact+cohesion)
More information20. A Dual-Primal FETI Method for solving Stokes/Navier-Stokes Equations
Fourteenth International Conference on Domain Decomposition Methods Editors: Ismael Herrera, David E. Keyes, Olof B. Widlund, Robert Yates c 23 DDM.org 2. A Dual-Primal FEI Method for solving Stokes/Navier-Stokes
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 informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationA recovery-assisted DG code for the compressible Navier-Stokes equations
A recovery-assisted DG code for the compressible Navier-Stokes equations January 6 th, 217 5 th International Workshop on High-Order CFD Methods Kissimmee, Florida Philip E. Johnson & Eric Johnsen Scientific
More information[abc] (2) b 2 =a 2 (a 2 b 1 ) b 1 b 1. b 1 (4) b 1 a 3 b 2 b 2 b 2
Problem 1. (a) Show that if Volumes and dual bases ɛ abc = g[abc] (1) Then show ɛ abc (defined from ɛ abc by raising indices, e.g. v a = g ab v b ) is 1 g [abc] (2) (b) Consider three vectors a 1, a 2,
More informationBasic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and Navier-Stokes equations
Basic hydrodynamics David Gurarie 1 Newtonian fluids: Euler and Navier-Stokes equations The basic hydrodynamic equations in the Eulerian form consist of conservation of mass, momentum and energy. We denote
More informationA finite difference Poisson solver for irregular geometries
ANZIAM J. 45 (E) ppc713 C728, 2004 C713 A finite difference Poisson solver for irregular geometries Z. Jomaa C. Macaskill (Received 8 August 2003, revised 21 January 2004) Abstract The motivation for this
More informationGraded Mesh Refinement and Error Estimates of Higher Order for DGFE-solutions of Elliptic Boundary Value Problems in Polygons
Graded Mesh Refinement and Error Estimates of Higher Order for DGFE-solutions of Elliptic Boundary Value Problems in Polygons Anna-Margarete Sändig, Miloslav Feistauer University Stuttgart, IANS Journées
More informationTermination criteria for inexact fixed point methods
Termination criteria for inexact fixed point methods Philipp Birken 1 October 1, 2013 1 Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany Department of Mathematics/Computer
More informationMACRO STOKES ELEMENTS ON QUADRILATERALS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 15, Number 4-5, Pages 729 745 c 2018 Institute for Scientific Computing and Information MACRO STOES ELEMENTS ON QUADRILATERALS MICHAEL NEILAN
More information