Discontinuous Galerkin methods Lecture 1 - Brown University

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sic understanding of the schemes through a simple example. The first DG schemes first schemes So let us consider the scalar problem e linear scalar wave equation ∂u ∂f(u) + =0, x ∈ [L, R] =Ω, ∂t ∂x near flux is given as f(u) =au. This is subject to the appropriate initial condit u(x, 0) = u0(x). onditions are given when the boundary is an inflow boundary, that is u(L, t) =g(t) if a ≥ 0, u(R, t) =g(t) if a ≤ 0. ate Ω by K nonoverlapping elements, x ∈ [x k l ,xk r]=D k , as illustrated in Fig. e elements we express the local solution as a polynomial of order N x ∈ D k : u k h(x, t) = Np n=1 û k n(t)ψn(x) = Np i=1 u k h(x k i ,t)ℓ k i (x).
sic understanding of the schemes through a simple example. The first DG schemes firstx schemes ∈ D k : u k k x − xk+1 h(x) =u xk x − xk + uk+1 − xk+1 xk+1 1 = u − xk k+i ℓ k i (x) ∈ Vh, So let us consider the scalar problem e linear scalar wave equation ise for the flux, f ∂u ∂f(u) + =0, x ∈ [L, R] =Ω, ∂t ∂x near flux is given as f(u) =au. This is subject to the appropriate initial condit We form the local residual u(x, 0) = u0(x). onditions are given when the boundary is an inflow boundary, that is k h . The space of basis functions is defined as Vh = ⊕K k k=1 ℓi of piecewise polynomial functions. Note in particular that there is no restric ss of the test functions between elements. the finite element case, we now assume that the local solution can be well re pproximation uh ∈ Vh and form the local residual x ∈ D k : Rh(x, t) = ∂uk h ∂t + ∂f k h − g(x, t), ∂x u(L, t) =g(t) if a ≥ 0, u(R, t) =g(t) if a ≤ 0. ate Ω by K nonoverlapping elements, x ∈ [xk l ,xkr]=D k lement. Going back to the finite element scheme, we recall that the global c ual are the source of the global nature of the operators M and S in Eq. (1.4). equire that the residual is orthogonal to all test functions φh ∈ Vh, leading t , as illustrated in Fig. e elements we express the local solution as a polynomial of order N x ∈ D k : u k h(x, t) = Np n=1 û k n(t)ψn(x) = Np i=1 i=0 u k h(x k i ,t)ℓ k i (x).
Page 1 and 2: RMMC 2008 Discontinuous Galerkin me
Page 3 and 4: A brief overview of what’s to com
Page 5 and 6: Lecture 1 • Why do we need someth
Page 7 and 8: What is causing this growing gap ?
Page 9 and 10: Can we close the gap ? It is the al
Page 11 and 12: Why high-order accuracy ? We may be
Page 13 and 14: U(x, 1.5 Why high-order accuracy ?
Page 15 and 16: We arrive at a more straightforward
Page 17 and 18: to advance the equations in time, t
Page 19 and 20: Finite difference schemes • Main
Page 21 and 22: evaluation of these fluxes is not s
Page 23 and 24: has to vanish for all φh ∈ Vh, t
Page 25: D So lets see how we can achieve th
Page 29 and 30: allow one to recover a meaningful g
Page 31 and 32: 1 Introduction The basics of DG-FEM
Page 33 and 34: ount of history A brief history ale
Page 35 and 36: A brief history • The last decade
Page 37: A brief overview - and what now ?

Discontinuous Galerkin methods Lecture 1 - Brown University

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