Discontinuous Galerkin methods Lecture 1 - Brown University

More documents

Recommendations

Info

volume methods (FVM), and finite element methods (FEM), as compared with the discontinuous Galerkin finite element method (DG-FEM)]. A represents success, Lets summarize the observations while ✕ indicates a short-coming in the method. Finally, a () reflects that the method, with modifications, is capable of solving such problems but remains a less natural choice. Complex High-order accuracy Explicit semi- Conservation Elliptic geometries and hp-adaptivity discrete form laws problems FDM ✕ FVM ✕ () FEM ✕ () DG-FEM () residual destroys the locality of the scheme and introduces potential problems with the stability for wave-dominated problems. On the other hand, this is precisely the regime where the finite volume method has several attractive features. An intelligent combination of the finite element and the finite volume methods, utilizing a space of basis and test functions that mimics the finite element method but satisfying the equation in a sense closer to the finite volume method, appears to offer many of the desired properties. This combination is exactly what leads to the discontinuous Galerkin finite element method (DG-FEM). To achieve this, we maintain the definition of elements as in the finite element scheme such that D k =[xk ,xk+1 What we need is a scheme that combines • The local high-order/flexible element of FEM • The local statement on the equation for FVM These are exactly the components of the Discontinuous Galerkin Finite Element Method ]. However, to ensure the locality
D So lets see how we can achieve this k = D k uv dx, u k D =(u, u) k D , oken inner product and norm h = K (u, v) k D , u k=1 2 Ω,h =(u, u) Ω,h . t Ω is only approximated by the union of D k , that is K Ω Ωh = D k=1 k 2.2 Basic elements of the sch D , stinguish the two domains unless needed. cal information as well as information from the neighintersection between two elements. Often we will refer tersections in an element as the trace of the element. cuss here, we will have two or more solutions or boundme physical location along the trace of the element. k Dk+1 Dk-1 x 1 l =L xk-1 xK r =xl k xk r =xl k+1 r =R Fig. 2.1. Sketch of the geometry for simple one-dimensional ex basis, ψn(x). A simple example of this could be ψn(x) =xn−1 . local grid points, xk i ∈ Dk Consider the linear scalar wave equation ∂u ∂f(u) + =0, x ∈ [L, R] =Ω, ∂t ∂x where the linear flux is given as f(u) =au. This is subject to the appro initial conditions u(x, 0) = u0(x). Boundary conditions are given when the boundary is an inflow bou that is u(L, t) =g(t) if a ≥ 0, u(R, t) =g(t) if a ≤ 0. We approximate Ω by K nonoverlapping elements, x ∈ [x , and express the polynomial through th k l ,xkr] = D illustrated in Fig. 2.1. On each of these elements we express the local so as a polynomial of order N = Np − 1 x ∈ D k : u k Np h(x, t) = û k Np n(t)ψn(x) = u k h(x k i ,t)ℓ k l r xpress the local solution as a polynomial of order N Np k k : uh(x, t) = û n=1 It is the multi-element component of FEM/FVM which gives the geometric flexibility i (x). k Np n(t)ψn(x) = u i=1 k h(x k i ,t)ℓ k i (x). two complementary expressions for the local solution. In the first one, we use a local polynomial basis, ψn(x). A simple example of this could lternative form, known as the nodal representation, we introduce Np = ∈ D k , and express the polynomial through the associated interpolating ). The connection between these two forms is through the definition of ûk n. We return to a discussion of these choices in much more detail later; e that we have chosen one of these representations. , t) is then assumed to be approximated by the piecewise N-th order uh(x, t), K u(x, t) uh(x, t) = u k=1 k and the solution is represented as h(x, t), native form, known as the nodal representation, we introduce N (x). The connection betwe forms is through the definition of the expansion coefficients, ûk i=1 n. W Here, we have introduced two complementary expressions for the loca a tion. discussion In the first of these one, known choices as in the much modal more form, detail we use later; a local for now polyn assume that we have chosen one of these representations. The global solution u(x, t) is then assumed to be approxim piecewise N-th order polynomial approximation uh(x, t), interpolating Lagrange polynomial, ℓk n=1 i with a high-order local basis as in FEM: • on modal form • on nodal form
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: has to vanish for all φh ∈ Vh, t
Page 27 and 28: sic understanding of the schemes 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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?