MA8502 Numerical solution of partial differential equations. The Poisson problem: Mixed Dirichlet/Neumann boundary conditions along curved boundaries

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1 MA85 Numerical solution of partial differential equations The Poisson problem: Mied Dirichlet/Neumann boundar conditions along curved boundaries Fall c Einar M. Rønquist Department of Mathematical Sciences NTNU, N-749 Trondheim, Norwa All rights reserved

2 Model problem ˆ Γ 4 Γ F ˆΓ 4 ˆΓ Γ ˆΓ Figure : The phsical domain = (, L ) (, L ) can be viewed as coming from the reference domain = (, ) (, ) through the one-to-one mapping F :. We consider here the numerical solution of the two-dimensional Poisson equation u = f in, with the following mied boundar conditions specified along : u Γ = { Dirichlet u n = u { Homogeneous Γ n = Neumann Γ4 conditions { u Non-homogeneous n = g() Γ Neumann Here, u n = u n is the derivative of u in the outward normal direction. We first formulate the corresponding weak formulation. Let X be the function space where X = {v H () v Γ = }. }{{} Onl Dirichlet conditions The weak form is then: Find u X such that u v d d = fv d d + ( u n)v ds = 4 k= Γ k u n v ds = ( u n) v ds v X, gv ds v X.

3 where We ma also formulate this problem abstractl as: Find u X such that a(u, v) = l(v) v X, a(u, v) = l(v) = u v d d, fv d d + gv ds. Γ } {{} Non-zero Neumann conditions We now proceed with a discretization based on high order polnomials. We first define the discrete space X N = {v X v F P()}. The discrete problem (with quadrature) is then: Find u N X N such that a N (u N, v) = l N (v) v X N, where a N (, ) and l N ( ) denote evaluation of the bilinear form and linear form b Gauss- Lobatto Legendre (GLL) quadrature. The net step is to derive a sstem of algebraic equations. To this end, we have to define a basis for our discrete space, X N : v X N, v F = ˆv, ˆv(ξ, η) = m= n= v mn l m (ξ) l n (η) where l m (ξ), m =,..., N are the usual one-dimensional Lagrangian interpolants through the GLL points. We remark that dim(x N ) = N(N + ). With this nodal, tensor-product basis, we can epress the numerical solution (in the reference variables) as Note that, since u Γ =, û N (ξ, η) = u mn l m (ξ)l n (η). () m= n= As usual, we choose test functions û N (ξ, ) = = u m =, m =,..., N. ˆv(ξ, η) = l i (ξ)l j (η) i =,..., N j =,..., N. ()

4 Using () and (), we are now read to give an eplicit epression for the bilinear form a N (u N, v). First, we note that ( L û N ˆv a(u N, v) = u N v d d = L ξ ξ + L ) û N ˆv dξ dη. L η η Using () and appling GLL quadrature, we arrive at (for this particular test function) with a N (u N, v) =  pq = m= n=  im ˆBjn + L ) ˆBim  jn u mn, L L ( L ρ α D αp D αq (the one-dimensional Laplace operator on (, )), α= ˆB pq = ρ p δ pq (the one-dimensional mass matri on (, )). In a similar fashion, we can derive an eplicit epression for the linear form l N (v). First, we note that l(v) = fv d d + gv ds = L L ˆf ˆv dξ dη + L ĝ ˆv dξ. 4 ˆ Again, using () and appling GLL quadrature, we arrive at l N (v) = L L 4 We note that α= β= ρ α ρ β f αβ l i (ξ α )l j (ξ β ) + L ρ α g α l i (ξ α )l j () α= = L L ρ i ρ j f ij + L 4 ρ ig i δ jn, i =,..., N, j =,..., N. }{{} onl contribution along J = L L 4 J S = L f ij = f((ξ i ), (ξ j )) g i = g((ξ i )) : Jacobian (volume) : Jacobian (surface) 4

5 ˆ Γ 4 Γ F ˆΓ 4 ˆΓ Γ ˆΓ Model problem The same problem as model problem, ecept that is now deformed. The same boundar conditions. The same abstract formulation. The domain is approimated using an isoparametric representation. The bilinear form a(u N, v) can now be epressed as a(u N, v) = ( ˆ ˆv) T G ˆ ûn dξ dη (see earlier notes), and a N (u N, v) is again obtained using GLL quadrature when evaluating the integral. The linear form l(v) is given as l(v) = fv d d + gv ds Γ = ˆf ˆvJ dξ dη + ĝ ˆvJ S dξ, ˆ where J = ξ J S = s ξ η η ξ Jacobian (volume), Jacobian (surface). s 5

6 Setting ˆv(ξ, η) = l i (ξ)l j (η) and appling GLL quadrature gives l N (v) = = α= β= α= β= ρ α ρ β J αβ f αβ l i (ξ α )l j (ξ β ) + ρ α ρ β J αβ f αβ δ iα δ jβ + ρ α Jα S g α l i (ξ α )l j () α= ρ α Jα S g α δ iα δ jn α= = ρ i ρ j J ij f ij + ρ i Ji S g i δ jn }{{} onl, i =,..., N, j =,..., N. contribution along Computationall, we let N P N (), N P N (), J = N N ξ η N η (( J S = ds dξ = N ξ N ξ, ˆΓ) + ( N ξ ) / ˆΓ). s ˆ Note also that f ij = f((ξ i, ξ j ), (ξ i, ξ j )), g i = g(s i ) = g(s(ξ i )), where (ξ i, ξ j ) = ij (ξ i, ξ j ) = ij } computed using Gordon-Hall s si coordinates ( in, in ) 6

7 Numerical results Let us conclude this section with some actual numerical results. We consider the solution of the Laplace equation in a 9 degree bend; see Figure. Γ 4 Γ Γ Figure : The computational domain corresponds to an aismmetric bend. Two of the edges correspond to a quarter of two concentric circles with radii r = and r =. We solve the Laplace equation (i.e., f = ) using a spectral approimation based on high order polnomials. The boundar conditions along Γ i, i =,,, 4 are as described in Model problem and Model problem. In order to approimate the geometr b high order polnomials, we first distribute N + GLL points along each of the four edges. The points are distributed radiall along Γ and Γ 4, and in an angular fashion along Γ and (i.e., according to arc length). Once the boundar points are distributed, we generate all the interior points using the Gordon-Hall algorithm described earlier. This will give us the phsical coordinates corresponding to all the GLL points on the reference domain = (, ). Figure and Figure 4 show the intermediate stages in this construction, as well as final set of points { mn } N m,n=. We can now compute N corresponding to an point (ξ, η) via our high order tensor-product basis N (ξ, η) = m= n= mn l m (ξ)l n (η). 7

8 Figure : The Gordon-Hall algorithm. Left plot: preserving Γ and (the mapping F ξ ). Right plot: preserving Γ and Γ 4 (the mapping F η ) Figure 4: The Gordon-Hall algorithm. Left plot: preserving the four corners (the mapping F ξη ). Right plot: the final mapping F = F ξ + F η F ξη. 8

9 We first impose a constant flu g = along the boundar. The numerical solution is depicted in Figure 5. As epected, we observe eponential convergence for this problem with regular data and geometr (represented using an isoparametric mapping) Error N Figure 5: Numerical results when imposing a constant flu g = along. Left plot: contour lines of the numerical solution, u N, to the Laplace problem. Right plot: the error u u N in the discrete L -norm as a function of the polnomial degree, N, used in the numerical approimation. Net, we impose a trigonometric variation in the heat flu g along the boundar. The numerical solution is depicted in Figure Figure 6: Numerical results when imposing a variable flu, g, along. Contour lines of the numerical solution, u N, to the Laplace problem. 9

10 Eercise Consider the curve Γ given b the function () = cos() for [, π]. We would like to compute the length L of this curve. The length can be epressed as L = ds, where ds is an infinitesimal length element along the curve Γ. Γ. Suggest a reparametrization of the curve on the form ((ξ), (ξ)), for ξ [, ]. Here, ξ = corresponds to the left end point of the curve, i.e. the point (, ), while ξ = corresponds to the right end point of the curve, i.e., the point (π, ).. Use the reparametrization to epress L as an integral with respect to ξ instead of s.. Compute the integral numericall using GLL quadrature.