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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

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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

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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

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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

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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 : γv = 0} and consider problem Find u ∈ X such that B(u, v) =F(v)

∀v ∈ X 0

(1a)

γu =g

(1b)

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Finite Element Approximation

A finite element approximation takes Xh ⊂ X and defines Mh = γXh ⊂ M and X 0

h = {v ∈ Xh : γv = 0} and solves: Find

uh ∈ Xh such that B(uh, v) =F(v)

∀v ∈ X 0

h

γuh =gh.

Here gh ∈ Mh approximates g. We assume the following discrete inf-sup condition holds

βuhX ≤

sup

v∈Xh,vX=1

B(uh, v) for all uh ∈ Xh.

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 5

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Error Estimate

A typical error estimate has the form

u − uhX ≤

  • 1 + κ

β

  • inf

wh∈Xh:γwh=gh u − whX,

Instead we hope to get an estimate of the form

u − uhX ≤ C( inf

vh∈Xh u − vhX + g − ghM),

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

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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 Lh : Mh → Xh such that

(P1) γLhµh = µh ∀µh ∈ Mh (P2) LhµhX ≤ CLµhM ∀µh ∈ Mh,

then the following error estimate holds

u − uhX ≤ C( inf

vh∈Xh u − vhX + g − ghM).

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 7

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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

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Our Result

We prove that such a discrete extension operator exists when Xh 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

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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

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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

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Raviart-Thomas space

Xh := {q ∈ H(div, Ω) : q|K ∈ [P0(K)]3+P0(K)x, for all K ∈ Th} Mh = {m ∈ L2(∂Ω) : m|F ∈ P0(F) for all faces F ⊂ ∂Ω}

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 12

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Harmonic extension

Given mh ∈ Mh with average zero define

−∆u =0

  • n Ω

∇u · n =mh

  • n ∂Ω

One has the following regularity result (see Dauge)

uH3/2+s(Ω) ≤ CmhHs(∂Ω)

for some s > 0. Also, one has

uH(div;Ω) ≤ CmhH−1/2(∂Ω)

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 13

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

We then define Lhmh = Π∇u. Where Π is the Raviart-Thomas projection.

  • Note that (P1) holds since (Π∇u) · n = mh on ∂Ω.
  • For (P2) to hold we need to show

Π∇uH(div;Ω) ≤ mhH−1/2(∂Ω)

To do that, one can use

Π∇uH(div;Ω) ≤Π∇u − ∇uH(div;Ω) + ∇uH(div;Ω) ≤C h1/2+suH3/2+s(Ω) + mhH−1/2(∂Ω) ≤C h1/2+smhHs(Ω) + mhH−1/2(∂Ω)

an inverse estimate gives h1/2+smhHs(Ω) ≤ CmhH−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

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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 ∈ Th we have

∇uH1/2+s(K) ≤(h−1/2−s

K

∇uL2(DK) + gHs(∂DK∩Γ) + h−s

K gL2(∂DK∩Γ)),

where DK := ∪{K′ ∈ Th : K ∩ K′ = ∅},

  • We localized inverse estimates (ala Ainsworth, McLean and

Tran). For any g ∈ Mh we have

  • F∈Γh

hFgh2

L2(F) gh2 H−1/2(Γ)

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 15

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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

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Nedelec space

Xh := {q ∈ H(curl, Ω) : q|K ∈ [P0(K)]3+[P0(K)]3×x, for all K ∈ Th} Mh ={m ∈ H(divΓ; ∂Ω) : m|F ∈ [P0(F)]2 + P0(F)xt for all faces F ⊂ ∂Ω} where xt is the tangential position vector. The trace norm is (see Buffa and Ciarlet)

mhM = mhH−1/2(∂Ω) + divΓmhH−1/2(∂Ω)

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 17

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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

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Let mh ∈ Mh then we know that divΓmh is then trace space of the Raviart-Thomas space. Hence, by previous result we have that there exists divvh = 0, vh · n = divΓmh

vhH(div;Ω) ≤ CdivΓmhH−1/2(∂Ω).

By exactness of the discrete De-Rham Complex there exists wh ∈ Xh (Nedelec space) such that curl(wh) = vh with

whH(curl;Ω) ≤ CvhH(div;Ω) ≤ CdivΓrhH−1/2(∂Ω).

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 19

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

If we define rh = mh − wh × n then we have divΓrh = divΓmh − divΓ(wh × n) = 0 By exactness of discrete de-Rham sequence (on surface mesh) we have rh = curlΓℓh with

ℓhH1/2(∂Ω) ≤ C rhH−1/2(∂Ω) ≤ CmhM

for some continuous piecewise linear ℓh. We can then find a piecewise linear uh on Ω such that uh is an extension of ℓh and

∇uhL2(Ω) ≤ CℓhH1/2(∂Ω).

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 20

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

We finally set Lhmh = wh + ∇uh Note that

(Lhmh) × n =wh × n + ∇uh × n =wh × n + curlΓℓh =wh × n + rh = mh LhmhH(curl;Ω) ≤ CwhH(curl;Ω) + C∇uhL2(Ω) ≤ mhM

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 21

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Final Remarks

  • Note that our technique for Raviart-Thomas used elliptic
  • regularity. It would be nice to obtain proof without using elliptic
  • regularity. In the two-dimensional case this has been done.
  • Not clear how to extend to higher dimensional De Rham

complexes.

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 22

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Happy Birthday Doug!!!

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 23

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

Argument in two dimensions

This result is due to Dominguez, Meddahi and Sayas (2007). Let Ω be a bounded connected polygon. Let mh be a piecewise constant on triangulation of ∂Ω with global average zero. Then we can find ℓh that is continuous piecewise linear such that

∂tℓh = mh

and one has

ℓhH1/2(∂Ω) ≤ CmhH−1/2(∂Ω).

Then define

−∆u =0

in Ω u =ℓh

  • n ∂Ω.
  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 24

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Motivation Our Contribution Techniques for Raviart-Thomas spaces Techniques for Nedelec spaces

continued...

with

uH1(Ω) ≤ CℓhH1/2(∂Ω) ≤ mhH−1/2(∂Ω).

Define Lhmh = curl(Ihu) where Ih is the Scott-Zhang interpolant Then Lhmh · n = curl(Ihu) · n = ∂tIhu = ∂tu = ∂tℓh = m. Also,

curl(Ihu)H(div;Ω) = curl(Ihu)L2(Ω) ≤ uH1(Ω) ≤ mhH−1/2(∂Ω).

  • J. Guzm´

an johnny_guzman@brown.edu Division of Applied Mathematics, Brown University. Discrete Extensions 25