DOLFIN User Manual - FEniCS Project

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DOLFIN User ManualHoffman, Jansson, Logg, Wells12.2 Convection–diffusionThe convection-diffusion equation with Dirichlet and homogeneous Neumannboundary conditions is given by:˙u + b · ∇u − ∇ · (a∇u) = f in Ω × (0, T ],u = g D on Γ 1 × (0, T ],−∂ n u = 0 on Γ 2 × (0, T ],u(·, 0) = u 0 in Ω,(12.3)where the convection is given by the vector b = b(x, t) and the diffusion isgiven by a = a(x, t).The variational formulation is:FIXME:Stabilized convection-diffusionThis is a stabilized FEM-formulation, so the solver can handle convectiondominatedproblems.The time integration is done using cG(1) (Crank-Nicolson).12.2.1 UsageThe API for the convection-diffusion solver:// Create convection-diffusion solverConvectionDiffusionSolver(Mesh& mesh, Function& w, Function& f,BoundaryCondition& bc);// Solve convection-diffusionvoid solve();80
DOLFIN User ManualHoffman, Jansson, Logg, Wells// Solve convection-diffusion (static version)static void solve(Mesh& mesh, Function& w, Function& f,BoundaryCondition& bc);A simple example of using the solver:int main(){dolfin_output("curses");Mesh mesh("dolfin.xml.gz");Convection w;Source f;MyBC bc;ConvectionDiffusionSolver::solve(mesh, w, f, bc);}return 0;12.2.2 PerformanceThere are no particular performance issues with the solver. GMRES is usedfor solving the linear system.12.2.3 LimitationsCurrently many coefficients (such as diffusivity) are not user-definable, theyneed to be exposed by the interface.81
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DOLFIN User ManualFebruary 24, 2006
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ContentsAbout this manual 91 Introd
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DOLFIN User ManualHoffman, Jansson,
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Chapter 3Linear algebra◮ Develope
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Page 29 and 30: Chapter 4The mesh◮ Developer’s
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Page 33 and 34: Chapter 5Functions◮ Developer’s
Page 43: Chapter 6Ordinary differential equa
Page 53 and 54: Chapter 8Nonlinear solver◮ Develo
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Page 65 and 66: Chapter 10The log systemDOLFIN prov
Page 85: Bibliography[1] Cygwin, 2005. http:
Page 88 and 89: v 2 v 0 = (0, 0)DOLFIN User ManualH
Page 105 and 106: Appendix CContributing codeIf you h
Page 109 and 110: Appendix DLicenseDOLFIN is licensed
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