ASPECTS OF CONVERGENCE FOR MIXED MULTISCALE FINITE ELEMENTS AND A NEW APPROACH TO THEIR DEFINITION Todd Arbogast James M. Rath Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin Supported provided in part by the U.S. National Science Foundation Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Second Order Elliptic PDE’S in Mixed Form Incompressible, single phase flow in a porous medium: u = − a ǫ ∇ p in Ω (Darcy’s law) ∇ · u = f in Ω (Conservation) u · ν = 0 on ∂ Ω (BC for simplicity) A mixed variational formulation: Find p ∈ W = L 2 / R and u ∈ V = H 0 (div) such that ( a − 1 u , v ) = − ( ∇ p, v ) = ( p, ∇ · v ) ∀ v ∈ V (Darcy’s law) ǫ ( ∇ · u , w ) = ( f, w ) ∀ w ∈ W (Conservation) Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Mixed Finite Element Approximation Find p ∈ W h ⊂ W and u ∈ V h ⊂ V such that ( a − 1 u h , v ) = ( p h , ∇ · v ) ∀ v ∈ V h ǫ ( ∇ · u h , w ) = ( f, w ) ∀ w ∈ W h Problem of scale: The coefficient a ǫ ( x ) varies on a fine scale ǫ ≪ 1. To resolve the solution, we need a mesh T h of maximal spacing h < ǫ . This is often not computationally feasible. Solution: We define V h × W h to respect the scales: • Multiscale finite elements (Babuˇ ska, Caloz & Osborn 1994; Hou & Wu 1997; Chen & Hou 2003) • Variational multiscale method (Hughes 1995, A., Minkoff & Keenan 1998, A. & Boyd 2006) Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Mixed Multiscale Finite Elements Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Preliminaries For this talk, • In all cases, W h = piecewise discontinuous constants • T h is a quasiuniform rectangular grid • E h are the mesh “edges” • For e ∈ E h , let E e be the two elements E e, 1 , E e, 2 ∈ T h bordering e E e, 1 E e, 2 e E e We consider multiscale finite elements defined either: • Elementwise on E ∈ T h • On dual-support domain E e for e ∈ E h . Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Raviart-Thomas Mixed FEM (RT)—1 We define v RT ∈ V RT for each coarse element edge e ∈ E h . e h Element definition: ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ��� ��� For each edge e ⊂ ∂E , solve ��� ��� ��� ��� ��� ��� v RT = −∇ φ RT ��� ��� in E, ��� ��� e e ��� ��� ��� ��� ∇ · v RT = ±| e | / | E | in E, ��� ��� e e ��� ��� e ��� ��� ��� ��� 0 on ∂E \ e, ��� ��� v RT ��� ��� · ν = ��� ��� e 1 on e, ��� ��� ����������������� ����������������� ����������������� ����������������� E e, 2 E e, 1 Dual-support definition (rectangular case): ����������������� ����������������� ����������������� ����������������� ��� ��� ����������������� ����������������� ��� ��� ��� ��� For each edge e ∈ E h , solve ��� ��� ��� ��� ��� ��� ��� ��� v RT = −∇ φ RT ��� ��� in E e , ��� ��� e e ��� ��� ��� ��� ∇ · v RT e ��� ��� = ±| e | / | E e,i | in E e,i , i = 1 , 2 , e ��� ��� ��� ��� v RT ��� ��� · ν = 0 on ∂E e . ��� ��� ����������������� ����������������� e ��� ��� ����������������� ����������������� ����������������� ����������������� E e Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Raviart-Thomas Mixed FEM (RT)—2 0.4 1.0 0.2 0.0 0.5 1.0 � 0.2 0.8 0.0 1.0 � 0.4 1.0 0.6 0.5 0.5 0.5 0.4 0.0 0.0 0.2 � 0.5 � 0.5 0.0 � 1.0 � 1.0 x -velocity y -velocity velocity 1.25 1 Theorem: (Raviart & Thomas, 1977) 0.75 0.5 � h � � u − u RT � 0 ≤ C � u � 1 h = O 0.25 h ǫ 0 0 0.25 0.5 0.75 1 normal trace Remark: These elements have no dependence on the scale ǫ . They are accurate only when h < ǫ , i.e., h resolves the fine-scale heterogeneity. Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Elements Based on the Heterogeneity Main idea of multiscale finite elements: In the boundary value problems used to define v RT ∈ V RT , insert the coefficient a ǫ ! e h Example: An permeability coefficient a ǫ 1.0 0.5 0.8 0.0 1.0 0.6 0.5 0.4 0.0 0.2 � 0.5 � 1.0 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Variational Multiscale Element (ME) Based on RT—1 We define v ME ∈ V ME for each coarse element edge e ∈ E h . e h Element definition: ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ��� ��� For each edge e ⊂ ∂E , solve ��� ��� ��� ��� ��� ��� v ME = − a ǫ ∇ φ ME ��� ��� in E, ��� ��� e e ��� ��� ��� ��� ∇ · v ME = ±| e | / | E | in E, ��� ��� e e ��� ��� e ��� ��� ��� ��� 0 on ∂E \ e, ��� ��� v ME ��� ��� · ν = ��� ��� e 1 on e, ��� ��� ����������������� ����������������� ����������������� ����������������� E e, 2 E e, 1 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Variational Multiscale Element (ME) Based on RT—2 1.5 0.4 0.2 1.0 0.0 1.0 0.5 � 0.2 0.8 0.0 1.0 � 0.4 1.0 0.6 0.5 0.5 0.5 0.4 0.0 0.0 0.2 � 0.5 � 0.5 0.0 � 1.0 � 1.0 x -velocity y -velocity velocity Theorem: (A. ’04; Chen & Hou ’03; A. & Boyd ’06) 1.5 1.25 1 � u − u ME � 0 ≤ C � u � 1 h, 0.75 h 0.5 � � � � u − u ME 0.25 � 0 ≤ C h � u 0 � 1 + ǫ � u 0 � 0 + ǫ/h � u 0 � 0 , ∞ , h 0 0 0.25 0.5 0.75 1 normal trace where u 0 is a smooth function independent of ǫ . � ǫ � h � �� � u − u ME � 0 = O min ǫ , h + ǫ + h h Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
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