Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces DISCRETE EXTENSION OPERATORS FOR MIXED FINITE ELEMENT SPACES ON LOCALLY REFINED MESHES Johnny Guzm´ an Division of Applied Mathematics, Brown University. October 23, 2014 Joint work with Mark Ainsworth(Brown University) and Francisco-Javier Sayas(University of Delaware) J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 1
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Outline 1 Motivation 2 Our Contribution 3 Techniques for Raviart-Thomas spaces 4 Techniques for Nedelec spaces J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 2
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Outline 1 Motivation 2 Our Contribution 3 Techniques for Raviart-Thomas spaces 4 Techniques for Nedelec spaces J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 3
Motivation Consider space X with trace space M where the trace operator γ : X → M is linear, continuous and surjective. Define X 0 = { v ∈ X : γ v = 0 } and consider problem Find u ∈ X such that ∀ v ∈ X 0 B ( u , v ) = F ( v ) (1a) γ u = g (1b)
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Finite Element Approximation A finite element approximation takes X h ⊂ X and defines M h = γ X h ⊂ M and X 0 h = { v ∈ X h : γ v = 0 } and solves: Find u h ∈ X h such that ∀ v ∈ X 0 B ( u h , v ) = F ( v ) h γ u h = g h . Here g h ∈ M h approximates g . We assume the following discrete inf-sup condition holds β � u h � X ≤ for all u h ∈ X h . B ( u h , v ) sup v ∈ X h , � v � X = 1 J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 5
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Error Estimate A typical error estimate has the form 1 + κ � � � u − u h � X ≤ w h ∈ X h : γ w h = g h � u − w h � X , inf β Instead we hope to get an estimate of the form � u − u h � X ≤ C ( inf v h ∈ X h � u − v h � X + � g − g h � M ) , Note that the approximation of g and u are separated . J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 6
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Discrete Lifting Dominguez and Sayas (2003) and Sayas (2007) show the following result. Theorem If there exists a uniformly bounded discrete extension operator L h : M h → X h such that ∀ µ h ∈ M h ( P 1 ) γ L h µ h = µ h ( P 2 ) � L h µ h � X ≤ C L � µ h � M ∀ µ h ∈ M h , then the following error estimate holds � u − u h � X ≤ C ( inf v h ∈ X h � u − v h � X + � g − g h � M ) . J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 7
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Outline 1 Motivation 2 Our Contribution 3 Techniques for Raviart-Thomas spaces 4 Techniques for Nedelec spaces J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 8
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Our Result We prove that such a discrete extension operator exists when X h is the Raviart-Thomas spaces or the Nedelec Spaces in three dimensions. We prove the result on general shape regular meshes that are not necessarily quasi-uniform. The domains we consider are connected, bounded polyhedral domains. J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 9
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Previous Results • In two dimensions for the Raviart-Thomas spaces the result was proved by Marquez, Meddahi and Sayas (2012). However, we do not see a way of extending their technique to three dimensions. • In three dimensions (also for Raviart-Thomas spaces) the results were proved by Babuska and Gatica (2003) and Gatica, Oyarzua and Sayas (2012). In both of these cases some quasi-uniformity is needed. J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 10
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Outline 1 Motivation 2 Our Contribution 3 Techniques for Raviart-Thomas spaces 4 Techniques for Nedelec spaces J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 11
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Raviart-Thomas space X h := { q ∈ H ( div , Ω) : q | K ∈ [ P 0 ( K )] 3 + P 0 ( K ) x , for all K ∈ T h } M h = { m ∈ L 2 ( ∂ Ω) : m | F ∈ P 0 ( F ) for all faces F ⊂ ∂ Ω } J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 12
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Harmonic extension Given m h ∈ M h with average zero define − ∆ u = 0 on Ω ∇ u · n = m h on ∂ Ω One has the following regularity result (see Dauge) � u � H 3 / 2 + s (Ω) ≤ C � m h � H s ( ∂ Ω) for some s > 0 . Also, one has � u � H ( div ;Ω) ≤ C � m h � H − 1 / 2 ( ∂ Ω) J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 13
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces We then define L h m h = Π ∇ u . Where Π is the Raviart-Thomas projection. • Note that (P1) holds since (Π ∇ u ) · n = m h on ∂ Ω . • For (P2) to hold we need to show � Π ∇ u � H ( div ;Ω) ≤ � m h � H − 1 / 2 ( ∂ Ω) To do that, one can use � Π ∇ u � H ( div ;Ω) ≤� Π ∇ u − ∇ u � H ( div ;Ω) + �∇ u � H ( div ;Ω) ≤ C h 1 / 2 + s � u � H 3 / 2 + s (Ω) + � m h � H − 1 / 2 ( ∂ Ω) ≤ C h 1 / 2 + s � m h � H s (Ω) + � m h � H − 1 / 2 ( ∂ Ω) an inverse estimate gives h 1 / 2 + s � m h � H s (Ω) ≤ C � m h � H − 1 / 2 ( ∂ Ω) . In this last step, previous analyses used quasi-uniformity of mesh. J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 14
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Adjustments to above proof for non quasi-uniform meshes • We use local regularity results instead of the global regularity result. For each K ∈ T h we have �∇ u � H 1 / 2 + s ( K ) ≤ ( h − 1 / 2 − s �∇ u � L 2 ( D K ) + � g � H s ( ∂ D K ∩ Γ) K + h − s K � g � L 2 ( ∂ D K ∩ Γ) ) , where D K := ∪{ K ′ ∈ T h : K ∩ K ′ � = ∅} , • We localized inverse estimates ( ala Ainsworth, McLean and Tran). For any g ∈ M h we have � h F � g h � 2 L 2 ( F ) � � g h � 2 H − 1 / 2 (Γ) F ∈ Γ h J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 15
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Outline 1 Motivation 2 Our Contribution 3 Techniques for Raviart-Thomas spaces 4 Techniques for Nedelec spaces J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 16
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Nedelec space X h := { q ∈ H ( curl , Ω) : q | K ∈ [ P 0 ( K )] 3 +[ P 0 ( K )] 3 × x , for all K ∈ T h } M h = { m ∈ H ( div Γ ; ∂ Ω) : m | F ∈ [ P 0 ( F )] 2 + P 0 ( F ) x t for all faces F ⊂ ∂ Ω } where x t is the tangential position vector. The trace norm is (see Buffa and Ciarlet) � m h � M = � m h � H − 1 / 2 ( ∂ Ω) + � div Γ m h � H − 1 / 2 ( ∂ Ω) J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 17
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Main idea Lagrange → Nedelec → Raviart-Thomas We already have discrete extensions for Raviart-Thomas and Lagrange spaces. J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 18
Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces Let m h ∈ M h then we know that div Γ m h is then trace space of the Raviart-Thomas space. Hence, by previous result we have that there exists div v h = 0 , v h · n = div Γ m h � v h � H ( div ;Ω) ≤ C � div Γ m h � H − 1 / 2 ( ∂ Ω) . By exactness of the discrete De-Rham Complex there exists w h ∈ X h (Nedelec space) such that curl ( w h ) = v h with � w h � H ( curl ;Ω) ≤ C � v h � H ( div ;Ω) ≤ C � div Γ r h � H − 1 / 2 ( ∂ Ω) . J. Guzm´ an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 19
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