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Finite Element Methods for the Advection-Diffusion Equation

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Finite Element Methods for the Advection-Diffusion Equation Powered By Docstoc
					Intro                CG Method               DG Method             C-DG Method   The Future




                 Finite Element Methods for the
                  Advection-Diffusion Equation

                                           John Chapman

                                 Mathematical Sciences, Durham University


                                            8th May 2008




John Chapman                                                                       Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method   C-DG Method   The Future




Outline

        Introduction

        The Continuous Galerkin Method

        The Discontinuous Galerkin Method

        The Continuous-Discontinuous Galerkin Method

        Work for the Future



John Chapman                                                      Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method           DG Method         C-DG Method   The Future

The Advection Diffusion Equation



The Advection Diffusion Equation


        We will be considering methods of solution for the model
        advection-diffusion (A-D) equation which is given by

                                  −ε∆u + b. u = f    in Ω ⊂ Rn
                                             u = g on ∂Ω

        with ε > 0 the diffusion coefficient, b ∈ [L2 (Ω)]n the velocity
        vector field and f, g ∈ L2 (Ω). We take Ω to be bounded and
        polygonal.




John Chapman                                                               Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method       C-DG Method        The Future

The Advection Diffusion Equation



Physical Interpretations and Applications



               Many relevant physical applications.
               These examples are not exhaustive.
               Two diverse problems to demonstrate the width of application.
               Examples drawn from “Numerical Solution of
               Convection-Diffusion Problems”, Morton (1996).




John Chapman                                                               Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method            DG Method          C-DG Method   The Future

The Advection Diffusion Equation



Pollutant dispersal in a river


        Consider a river surface as a two dimensional domain which
        contains a depth averaged pollutant concentration c. Then the
        steady state representation of its dispersal may be given by

                                      v. c =    .(D c) + S

        where S is the source of the pollutant, v a horizontal velocity field
        and D a diffusivity coefficient. This problem is representative of
        advection diffusion problem if D is independent of c.




John Chapman                                                                   Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method               DG Method    C-DG Method       The Future

The Advection Diffusion Equation



Semi-conductor equations


        Two parts of the basic model for the steady state distribution of
        electrons in a semi-conductor are
                                      qR =    .Jn
                                      Jn = qµn (Ut n − n ψ)

        Clearly a substitution of the second equation into the first results
        in a advection-diffusion problem for n.




John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                  CG Method                DG Method        C-DG Method   The Future

Analytic Solutions and Behaviour



Analytic Solution of A-D Eqns.
        In simple cases it is possible to solve our model A-D equations
        analytically using asymptotic expansions or the method of
        characteristics. We won’t go into detail here, but use some results
        to further our understanding of the behaviour of the equations.
        Let us consider the simple equation

                         −εu + u = 1 in (0, 1),             u(0) = u(1) = 0

        which has the solution
                                                exp(− 1−x ) − exp(− 1 )
                                                       ε            ε
                                   u(x) = x −
                                                    1 − exp(− 1 )
                                                               ε




John Chapman                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                  CG Method      DG Method         C-DG Method            The Future

Analytic Solutions and Behaviour



Boundary Layers
        Plotting this equation for several values of ε we see the onset of a
        boundary layer.




            Figure: Advection Diffusion Boundary Layer for ε = 1.0, 0.1, 0.01



John Chapman                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                 CG Method       DG Method         C-DG Method         The Future

General Finite Element Methods



A Short Introduction to Galerkin Finite
Element Methods
        The emphasis here is on short, with details being expanded in the
        next section.
               Find the weak formulation of the problem.
               Replace the infinite dimensional space with a finite
               dimensional subspace, resulting in a problem we can solve.
               Take successively larger subspaces and show that the solution
               converges to the solution of the original problem.
               This is the Galerkin method.
               If the subspace is of piecewise polynomials on a triangulation
               of the region, we call it a Finite Element Method.

John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                DG Method         C-DG Method   The Future

CG FEMs for the Advection Diffusion Equation



Formulation of the Problem

        Returning to our model problem

                                  −ε∆u + b. u = f         in Ω ⊂ Rn
                                                  u = 0 on ∂Ω

        with g = 0 we follow the steps of the previous slide. The weak
        formulation is
                                                                1
               Bε (u, v) = ε( u, v) + (b. u, v) = (f, v) ∀ v ∈ H0 (Ω)

        with ( , ) being the usual L2 inner product.



John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                 DG Method                C-DG Method   The Future

CG FEMs for the Advection Diffusion Equation



Existence and Uniqueness of Solution


        Provided we can show

                            |Bε (u, v)| ≤ α u        1
                                                    H0     v      1
                                                                 H0   continuity,

                                                           2
                                      Bε (u, u) ≥ β u      H01    coercivity

        and have f ∈ V , existence and uniqueness follow from the
        Lax-Milgram theorem.




John Chapman                                                                            Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                DG Method          C-DG Method   The Future

CG FEMs for the Advection Diffusion Equation



Finite Element Approximation

        Let T be a shape regular triangulation of Ω. We consider a space

                Vh := {v ∈ H 1 (Ω) : v|T ∈ P k (T ), v|∂Ω = 0, ∀T ∈ T },

        where P k (T ) is the space of polynomials of degree at most k
        supported in T . Then the continuous Galerkin finite element
        approximation to the A-D equation is: Find uh ∈ Vh such that

                                      Bε (uh , v) = (f, v) ∀v ∈ Vh .

        and existence and uniqueness again follows from Lax-Milgram.



John Chapman                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                  DG Method            C-DG Method            The Future

CG FEMs for the Advection Diffusion Equation



Stability of CG
        We want to demonstrate that this finite element approximation has
        not got good error characteristics as ε → 0.
                         e
        We derive from C´a’s lemma

                                              (ε2 + b   2 )1
                                2                       ∞ 2                        2
                   u − uh         1
                                H0 (Ω)   ≤                     (1 + c) u − vh        1
                                                                                   H0 (Ω)
                                                   ε
        which indicates a problem as
                                                                                    1
                       (ε2 + b        2 )1                          b 2             2
                                      ∞ 2
                                              (1 + c) = (1 + c) 1 + 2∞
                            ε                                       ε
                                                      → ∞ as ε → 0.


John Chapman                                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                DG Method   C-DG Method       The Future

CG FEMs for the Advection Diffusion Equation



Behaviour of Numerical Solutions




               Figure: Instability in CG Method (Ern and Guermond, p.166)


John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method        C-DG Method          The Future




The Discontinuous Galerkin Method



               Clearly the application of a continuous Galerkin method is
               limited.
               One alternative is the discontinuous Galerkin method.
               We replace discontinuous test space with a discontinuous one.
               There are other solutions too, but we do not cover them here.




John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                 CG Method                   DG Method   C-DG Method   The Future

History and Development of the Discontinuous Method



Initial Work


               Introduced by Reed and Hill (1973) for solving the Neutron
               Transport Equation.
               Analysis followed, notably by LeSaint and Raviart (1974) who
               proved a rate of convergence of (∆x)k .
               This rate has been improved upon, with Peterson (1991)
               proving an optimal order in the general case.
               Better results have been proven for the case of special meshes
               (e.g. Richter (1988)).




John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                 CG Method                   DG Method   C-DG Method    The Future

History and Development of the Discontinuous Method



Extension to the A-D Equations

               Richter (1992) presented a extension to the linear A-D
               equations, including an optimal order of convergence in the
               advection dominated case.
               Bassi and Rebay (1997) formulated a method of DG space
               discretisation for the compressible Navier Stokes equations
               (related to this problem).
               Cockburn and Shu (1998) introduced the Local Discontinuous
               Galerkin (LDG) method, rewriting the original system as a
               larger first order system which is then solved using DG
               methods.



John Chapman                                                                   Durham
FEM for the Advection-Diffusion Eqn.
Intro                 CG Method                   DG Method   C-DG Method   The Future

History and Development of the Discontinuous Method




                                     u
               Houston, Schwab and S¨li (2000) considered an extension to
               the method that does not rely on streamline stabilization.
               Buffa, Hughes and Sangalli (2006) presented a new approach
               to the stability of the DG method for the A-D equation, which
               we will look at in more detail later.




John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diffusion Equation



Formulation of the Problem

        The test space is now discontinuous

                          Vh := {v ∈ L2 (Ω) : v|T ∈ P k (T ) ∀T ∈ T }

        and we introduce the notation for averages and jumps across edges
                               1
                          {v} = (v + + v − ) [v ] = v + n+ + v − n−
                               2
                              1
                         {v} = (v+ + v− ) [v] = v+ n+ + v− n−
                              2



John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method        C-DG Method   The Future

The DG Method for the Advection Diffusion Equation



Elements and Normals

                                                    T+
                                                                n-
                                       T-


                                                n+


                                      Figure: Normals and ± Regions

John Chapman                                                                         Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diffusion Equation



More Formulation of the Problem

        Call the boundary of the region ∂Ω = Γ and take the inflow and
        outflow boundaries Γ− , Γ+ with

                                      Γ− = {x ∈ Γ : b(x).n(x) ≤ 0}
                                      Γ+ = {x ∈ Γ : b(x).n(x) > 0} .

        For a triangulation T the skeleton Eh can be decomposed into Eh ,o

        the internal edges and Γ for the external edges (abusing notation).
        For each T ∈ T we can similarly denote the inflow and outflow
        edges by ΓT .



John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method        C-DG Method           The Future

The DG Method for the Advection Diffusion Equation



Representation of Region and Element

                      Γ-                 Γ+



                                                                Γ-          T          +
                                                                T
                                                                                   Γ
                                                                                   T


                                                                                       b




                         Figure: Boundary of Region Ω and Element T


John Chapman                                                                                 Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diffusion Equation




        There are different presentations of the DG method corresponding
        to different penalizations of the use of discontinuous spaces. Here
        we the the symmetric interior penalty method (SIPG), but other
        approaches are just as valid. See Arnold et al. (2001) for a
        thorough approach to different methods (applied to the elliptic
        equation).




John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                         DG Method                  C-DG Method            The Future

The DG Method for the Advection Diffusion Equation



The DG Method
        Define the bilinear forms

         Bd (u, v) =                        u. v −                     { u.n}[v ] + { v.n}[u]
                          T ∈T        T                      o
                                                          e∈Eh     e


                              −                 ( u.n)v + ( v.n)u +                           σh−1 [u].[v ]
                                                                                                ⊥
                                  e∈Γ       e                                      e∈Eh   e


         Ba (u, v) =                      (b. u)v −                        (b.n)[u]v +
                                      T                                e
                          T ∈T                             e∈Γ−
                                                              T


                              −                     (b.n)u+ v +
                                                e
                                  e∈Γ−
                 Bε = εBd + Ba

John Chapman                                                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method    C-DG Method   The Future

The DG Method for the Advection Diffusion Equation



The DG method




        The discontinuous Galerkin method is: Find uh ∈ Vh such that

                                      Bε (uh , v) = (f, v) ∀v ∈ Vh .




John Chapman                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method                  C-DG Method   The Future

The DG Method for the Advection Diffusion Equation



A Note on Norms


        The norms used are
                                          2
                                      v   d   = Bd (v, v)
                                                    1
                                      v   2
                                          a   =       ( .b)v 2 +              |b.n| [v ]2
                                                 Ω 2                     Eh
                                      2               2           2
                                  v   DG      =ε v    d   + v     a
                                      2              2                             2
                              v       SDG     = v    DG   +            τT b. v     T
                                                                T ∈T




John Chapman                                                                                   Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method   C-DG Method           The Future

The DG Method for the Advection Diffusion Equation



Stability Using the Method of Buffa, Hughes
and Sangalli

        Theorem
        There exists σ such that for all σ ≥ σ ,
                     ˆ                       ˆ
                                                  Bε (v, w)
                                 inf sup                    ≥ζ>0
                               v∈Vh w∈Vh        v SDG w SDG

        where ζ is independent of h,ε,b and the domain.

        Proof.
        Show w SDG       v SDG and Bε (v, w) ≥ β v                 2      using the
                                                                   SDG
        method of Buffa, Hughes and Sangalli (2006).


John Chapman                                                                            Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                      DG Method   C-DG Method          The Future

The DG Method for the Advection Diffusion Equation



Why inf sup?



                                ˜           ∗
               Consider the map B : Vh → Vh .
                                       ˜
               What can one say about B −1 ?
                                                              ˜
               In fact, the inf sup constant gives a bound on B −1            Vh →Vh ,
                                                                               ∗

               provided it is independent of ε.
               This implies stability.




John Chapman                                                                             Durham
FEM for the Advection-Diffusion Eqn.
Intro                 CG Method          DG Method           C-DG Method            The Future

Limitations of the Methods



Advantages and Limitations of the DG
Method

               Pros
                      Can be made arbitrarily accurate by increasing the order of
                      approximating polynomial element by element.
                      Highly parallelizable.
                      Suitable for complex geometries.
                      Easy to apply mesh refinement due to lack of continuity.
               Cons
                      Increase of degrees of freedom over CG methods.
                      More costly to compute (in general).




John Chapman                                                                          Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method   C-DG Method       The Future

CDG Method



Characteristics of a New Method



               Global stability.
               Ease of implementation.
               Low cost.
        We consider the work started by Max Jensen, Andrea Cangiani and
        Emmanuil Georgoulis in 2006.




John Chapman                                                          Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method        C-DG Method        The Future

CDG Method



Description of the Problem



        We look for a method that implements
               CG away from boundary layers in a region ΩC (low cost).
               DG around the boundary layer in a region ΩD (high stability).
               New behaviour at the junction of the CG and DG regions.
        We look at each of these points separately.




John Chapman                                                                Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                                                  DG Method        C-DG Method   The Future

CDG Method


                                                                  Ω


                                                                                            ΩD
                       Assumptions for unperturbed region valid




                                                                             ΩC




                                                                  Figure: Partition into CD and DG Regions

John Chapman                                                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method             DG Method         C-DG Method   The Future

CDG Method



Finding a CG region

        Consider again the original A-D equation, with ε small. Here we
        want the solution u to be close to a solution uC of an unperturbed
        problem, such as

                                      b. uC =f    in ΩC ⊂ Rn ,
                                        uC =0 on ∂ΩC− .

        We can formalize this relationship and prove that for a solution
        u = uC + uD then u → uC in H 1 (ΩC ) as ε → 0. The assumptions
        made to show this result are reasonable for smooth domains away
        from boundary layers.



John Chapman                                                                 Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method          DG Method           C-DG Method       The Future

CDG Method



Stability on the CG region

        Theorem
        Let uε,h be the DG approximation to u = uC + uD on Vh . Assume
        uC ∈ H 2 (ΩC ). Assume k ≥ 1. Assume shape-regularity. Assume
        − .b ≥ γ > 0 on ΩC . Then
                                                    √                1
                    |u − uε,h |H 1 (ΩC ) ≤ C(f )(k 2 ε + k 4 + k 2 h 2 + h)


        Proof.
        Show the result on uC and uD , then use linearity of the method to
                                                 1
        extend to u. This holds provided h ∈ (1, 2 ] and ε ∈ [0, 1].


John Chapman                                                                    Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method             DG Method        C-DG Method     The Future

CDG Method



Finding a DG region


        On the region ΩD we can use the results of Buffa, Hughes and
        Sangalli, as defined previously. We take a space of test functions

                    Vh :={v ∈ L2 (Ω) : v|T ∈ P k (T ),
                                 v|∂Ω∪∂ΩC = 0, v|ΩC ∈ H 1 (ΩC ) ∀T ∈ T }.

        Notice that this space means that on ΩC the usual continuous
        Galerkin method will be implemented. We have now partitioned
        our region as shown on a previous slide.




John Chapman                                                                  Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method         DG Method            C-DG Method        The Future

CDG Method



Region Interaction


               It’s not clear if extra proof is needed to resolve the region
               boundary.
               We have shown that the DG method works on the CG region
               and on the DG region.
               I present the current, not final, thinking.
               Take the streamline diffusion parameter τ to be switched off
               in the boundary region.
               Consider the solution across critical edges.




John Chapman                                                                     Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method                     DG Method                   C-DG Method   The Future

CDG Method




                                                No man's land

                              ΩC                                                        ΩD




                                                      τ=0                      τ≠0




                                                               Critical edge
                    Assumptions for unperturbed region valid




                                       Figure: Element Boundary



John Chapman                                                                                   Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method        C-DG Method         The Future

CDG Method



Numerical Experiments




        Results of numerical experiments performed by Cangiani,
        Georgoulis and Jensen (2006).
        The results are for a 10 × 10 grid with Ω = (0, 1)2 , ε = 10−2 ,
        b = (1, 1)T , f = 1.




John Chapman                                                                 Durham
FEM for the Advection-Diffusion Eqn.
Intro                   CG Method                          DG Method                              C-DG Method                                        The Future

CDG Method




             2                                                                   2


            1.5                                                                 1.5


             1                                                                   1


            0.5                                                             1   0.5                                                              1
                                                                      0.8                                                                  0.8
                                                                0.6                                                                  0.6
             0                                            0.4                    0                                             0.4
             0    0.2                                                            0    0.2
                         0.4                        0.2                                     0.4                          0.2
                               0.6    0.8                                                            0.6   0.8
                                            1   0                                                                1   0




        Figure: Comparison of DG (left) and CDG (right) Methods with f=1,
        ε = 10−2 on a 10x10 grid


John Chapman                                                                                                                                           Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method             DG Method   C-DG Method        The Future

The Future



Stability Across the Whole Region

        Conjecture
        There exists σ such that for all σ ≥ σ ,
                     ˆ                       ˆ
                                             Bε (v, w)
                                 inf sup               ≥ζ>0
                               v∈Vh w∈Vh   v SDG w SDG

        where ζ is independent of h,ε,b and the domain.

        Idea for Proof
        Follow the method as for DG and adapt it to handle the critical
        edges. The difficult question is how to turn τ off.



John Chapman                                                                Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method         DG Method         C-DG Method   The Future

The Future



Complete Stability Analysis




               Complete proof above.
               Incorporate the second join into the stability proof.
               Perform some numerical experiments.




John Chapman                                                             Durham
FEM for the Advection-Diffusion Eqn.
Intro                CG Method        DG Method        C-DG Method     The Future

The Future



Conclusions



               The inadequacy of the CG method for the advection-diffusion
               equation.
               The development of stable DG methods.
               Search for a suitable combined method with favourable
               characteristics of both CG and DG.




John Chapman                                                                Durham
FEM for the Advection-Diffusion Eqn.

				
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