Discontinuous Galerkin methods Lecture 3 - Brown University

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where To summarize ℓ =[ℓ1(r),...,ℓNp matters, we have local approximations of the form (r)]T and ˜ P (r) =[ ˜ P0(r),..., ˜ PN (r)] T . W ested in the particular solution, ℓ, which minimizes the Lebesque This highlights that it is the overall structure of the nodes rather than the details of the individual node position that is important; for example, one could optimize these nodal sets Npfor various applications. Np Lets summarize this u(r) uh(r) = ûn ˜ we recall Cramer’s rule for solving linear systems of equations Pn−1(r) = u(ri)ℓi(r), (3.3 So we have the local approximations n=1 To summarize matters, we have local approximations i=1 of the form ℓi(r) = Det[VT (:, 1), VT (:, 2), . . . , ˜ P (r), VT (:,i+ 1),...,VT (: Det(VT ) where ξi = ri are the Legendre-Gauss-Lobatto quadrature points. A centr Np u(r) uh(r) = ûn ˜ Np component of this construction is thePn−1(r) Vandermonde = u(ri)ℓi(r), matrix, V, which (3.3) estab lishes the connections n=1 i=1 It suggests that it is reasonable to seek ξi such that the denomina where ξi = ri are the Legendre-Gauss-Lobatto quadrature points. A central component of this u = construction Vû, V is the Vandermonde matrix, V, which establishes the connections T ℓ(r) = ˜ P (r), Vij = ˜ determinant of V), is maximized. Pj(ri). For this one dimensional case, the solution to this problem is relatively simple form as the Np zeros of [151, 159] and are the Legendre Gauss Lobatto points: u = Vû, V T ℓ(r) = ˜ P (r), Vij = ˜ By carefully choosing the orthonormal Legendre basis, Pj(ri). ˜ ri Pn(r), and the nod points, ri, we have ensured that V is a well-conditioned object and that th resulting interpolation zeros of is well behaved. A script for initializing V is given i Vandermonde1D.m. By carefully choosing the orthonormal Legendre basis, ˜ f(r) = (1 − r Pn(r), and the nodal points, ri, we have ensured that V is a well-conditioned object and that the resulting interpolation is well behaved. A script for initializing V is given in Vandermonde1D.m. 2 ) ˜ P ′ N (r). These This areleads closely to a related robust toway theof normalized computing/evaluating Legendre polynomi known a high-order as the Legendre-Gauss-Lobatto polynomial approximation. (LGL) quadrature points library routine JacobiGL.m in Appendix A, these nodes can be co but is it accurate ? >> [r] = JacobiGL(0,0,N);
e notation A second look at approximation k on a more rigorous discussion, we need to introduce some ion. InWe particular, will need both global a little norms, more defined notation on Ω, and broed over Ωh as sums of K elements D Regular energy norms k , need to be introduced. ion, u(x,t), we define the global L2-norm over the domain u 2 Ω = u Ω 2 dx oken norms u 2 K Ω,h = u k=1 2 k D , u2 k D = D k u 2 dx. we define the associated Sobolev norms u (α) 2 Ω, u 2 K Ω,q,h = u 2 D k , u2 ,q D k ,q = q u (α) 2 additional notation. In particular, both global norms, defined on Ω, a ken norms, defined over Ωh as sums of K elements D k , need to be intr For the solution, u(x,t), we define the global L2-norm over the Ω ∈ R d as u 2 Ω = u Ω 2 dx as well as the broken norms u 2 K Ω,h = u k=1 2 k D , u2 k D = D k u 2 dx. In a similar way, we define the associated Sobolev norms u 2 q Ω,q = u |α|=0 (α) 2 Ω, u 2 K Ω,q,h = u k=1 2 D k , u2 ,q D k ,q = q u |α|=0 where α is a multi-index of length d. For the one-dimensional cas often considered in this chapter, this norm is simply the L2 Before we embark on a more rigorous discussion, we need to introduce some additional notation. In particular, both global norms, defined on Ω, and broken norms, defined over Ωh as sums of K elements D -norm of k , need to be introduced. For the solution, u(x,t), we define the global L2-norm over the domain Ω ∈ R d as u 2 Ω = u Ω 2 dx as well as the broken norms u 2 K Ω,h = u k=1 2 k D , u2 k D = D k u 2 dx. In a similar way, we define the associated Sobolev norms u 2 q Ω,q = u |α|=0 (α) 2 Ω, u 2 K Ω,q,h = u k=1 2 D k , u2 ,q D k ,q = q u |α|=0 (α) 2 D k, where α is a multi-index of length d. For the one-dimensional case, most often considered in this chapter, this norm is simply the L2 Sobolev norms 76 4 Insight through theory derivative. We will also define the space of functions, u ∈ H Semi-norms -norm of the q-th q (Ω), as those functions for which uΩ,q or uΩ,q,h is bounded. Finally, we will need the semi-norms k=1 |u| 2 Ω,q,h = |α|=0 D k ,q D k, u (α) 2 D k. ulti-index of length d. For the one-dimensional case, most in this chapter, this norm is simply the L2-norm of the q-th 4.2 Briefly on convergence K k=1 |u| 2 D k ,q , |u|2 = |α|=q
Page 1 and 2: RMMC 2008 Discontinuous Galerkin me
Page 3 and 4: Lecture 3 • Let’s briefly recal
Page 5 and 6: Let us recall We already know a lot
Page 7 and 8: ∂t ∂x ∂y assume approximation
Page 9 and 10: d dt b u dx = d (λt − a)u −
Page 11 and 12: which one can attempt to express us
Page 13 and 14: x ∈ D Lets move on At this point
Page 15 and 16: mial basis). This leaves only the q
Page 17 and 18: erty quadrature ˜Pn(r) of wesuch u
Page 19 and 20: It suggests that it is reasonable t
Page 21: Local approximation - an example De
Page 25 and 26: simplicity, we assume that all elem
Page 27 and 28: (N +1+σ)! n=N+1 v − vh Approxima
Page 29 and 30: h h This implies vh(r) = ˜vh(r)+
Page 31 and 32: the situation is different. Note th
Page 33: Lets summarize Fluxes: • For line

Discontinuous Galerkin methods Lecture 3 - Brown University

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