UFL Specification and User Manual 0.3 - FEniCS Project

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggnumber of points for quadrature elements, and let the form compiler choosea quadrature rule. This way the form depends less on the cell in use.WeconsiderhereanonlinearversionofthePoisson’sequationtoillustratethemain point of the "Quadrature" finite element family. The strong equationlooks as follows:−∇·(1+u 2 )∇u = f. (3.23)The linearised bilinear and linear forms for this equation,a(v,u;u 0 ) =L(v;u 0 ,f) =∫∫ΩΩ∫(1+u 2 0)∇v ·∇udx+ 2u 0 u∇v ·∇u 0 dx, (3.24)Ω∫vf dx− (1+u 2 0)∇v ·∇u 0 dx, (3.25)can be implemented in a single form file as follows:Ω# NonlinearPoisson.uflelement = FiniteElement("Lagrange", triangle, 1)v = TestFunction(element)u = TrialFunction(element)u0 = Function(element)f = Function(element)a = (1+u0**2)*dot(grad(v), grad(u))*dx \+ 2*u0*u*dot(grad(v), grad(u0))*dxL = v*f*dx - (1+u0**2)*dot(grad(v), grad(u0))*dxHere, u 0 representsthesolutionfromthepreviousNewton-Raphsoniteration.The above form will be denoted REF1 and serve as our reference implementationfor linear elements. A similar form (REF2) using quadratic elementswill serve as a reference for quadratic elements.Now, assume that we want to treat the quantities C = (1 + u 2 0) and σ 0 =(1+u 2 0)∇u 0 as given functions (to be computed elsewhere). Substituting into71