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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.