Automated Modeling with FEniCS

Transcription

1 Automated Modeling with FEniCS L. Ridgway Scott The Institute for Biophysical Dynamics, The Computation Institute, and the Departments of Computer Science and Mathematics, The University of Chicago UChicago September /65

2 Modeling with PDEs Partial differential equations (PDEs) used pervasively to model phenomena of interest Analytical techniques for solving PDEs limited in scope able to solve only the simplest model problems. Development of reliable numerical methods facilitate PDE solution Modern computers and software techniques help Most widely used technique is finite element method (FEM) FEniCS Project automates generation of FEM software UChicago September /65

3 Rethinking courses PDE s applied: separation of variables theory: requires measure theory (Lions Red Book, 1969) Numerical approximation applied: finite differences, simple PDEs theory: Sobolev spaces with lots of indices (Babuska and Aziz, Maryland Conference, 1972) UChicago September /65

4 New goal A lot of theory has been developed in 50 years So why can t we Combine theory for both PDE s and numerical methods Use software that automates generation of PDE/FEM software Focus on description of theory and critical PDE problems PDE models as variables (new materials) UChicago September /65

5 Variational formulations The finite element method is based on the variational formulation of partial differential equations (PDEs). The variational formulation has two advantages: It provides a language to define PDEs suitable for compilation into executable code It provides a basis for a theory of PDEs that allows one to know whether or not a given model is well posed. We explain both these points via a simple example, Laplace s equation. Subsequently, a large variety of problems will be shown to fit into this framework. UChicago September /65

6 Laplace equation Define a function u in a domain Ω R d by together with boundary conditions u = f in Ω (1) u = 0 on Γ Ω (Dirichlet) u n = 0 on Ω\Γ (Neumann) (2) where u n denotes the derivative of u in the direction normal to the boundary, Ω ( u n = n u.) This is known variously as Poisson s equation or Laplace s equation (especially when f 0). UChicago September /65

7 Laplace applications This equation forms the basis of a remarkable number of physical models. It serves as a basic equation for diffusion, elasticity, electrostatics, gravitation, and many more domains. In potential flow, the gradient of u is the velocity of incompressible, inviscid, irrotational fluid flow. The boundary Ω is the surface of an obstacle moving in the flow, and one solves the equation on the exterior of Ω. UChicago September /65

8 Solution space Right place to look for solution of such an equation is a Sobolev space denoted H 1 (Ω) defined by H 1 (Ω) = { v L 2 (Ω) : v L 2 (Ω) d}, (3) where L 2 (Ω) means functions square integrable on Ω: ( 1/2 v L2 (Ω) = v(x) dx) 2 <, Ω L 2 (Ω) d means d copies of L 2 (Ω) (Cartesian product). There is a natural inner-product on L 2 (Ω) defined by (v,w) L2 (Ω) = v(x) w(x) dx, (4) Ω UChicago September /65

9 Functions spaces Inner-product and associated norm on H 1 (Ω) defined by (v,w) H1 (Ω) = (v,w) L2 (Ω) + v(x) w(x)dx v H1 (Ω) = UChicago September /65 Ω (v,v) H1 (Ω) For a vector-valued function w, e.g., w = v, we define ( 1/2, w L2 (Ω) = w L2 (Ω) = w(x) dx) 2 where ξ denotes Euclidean norm of vector ξ R d, and (v,w) L2 (Ω) = v(x) w(x)dx. Ω Ω

10 Domain for Poisson s Equation Ω (a) (b) Figure 1: (a) Domain Ω with Γ indicated in red. (b) Triangulation of Ω. Now consider equation (1) with boundary conditions (2). Assume Γ has nonzero measure (length or area, or even volume, depending on dimension). Later, we will return to the case when Γ is empty, the pure Neumann case. Typical Ω shown in Figure 1, with Γ shown in red. UChicago September /65

11 Boundary condtions for Poisson s Equation To formulate the variational equivalent of (1) with boundary conditions (2), we define a variational space that incorporates the essential, i.e., Dirichlet, part of the boundary conditions in (2): V := { v H 1 (Ω) : v Γ = 0 }. (5) See Table 1 for an explanation of the various names used to describe different boundary conditions. generic name example honorific name essential u = 0 Dirichlet u natural n Neumann Table 1: Nomenclature for different types of boundary conditions. UChicago September /65

12 Variational Formulation of Poisson s Equation The appropriate bilinear form for the variational problem is determined by multiplying Poisson s equation by a suitably smooth function, integrating over Ω and then integrating by parts: (f,v) L2 (Ω) = ( u)v dx = = Ω Ω Ω u vdx UChicago September /65 Ω v u n ds u vdx := a(u,v). The boundary term in (6) vanishes for v V because either v or u n is zero on any part of the boundary. (6)

13 Integration by parts The integration-by-parts formula derives from the divergence theorem w(x)dx = w(s) n(s)ds (7) Ω applied to w = v u, together with u n = ( u) n and u = ( u) (n is the outward-directed normal to Ω). More precisely, we observe that (v u) = = d i=1 UChicago September /65 Ω ( d (v u)i ),i = (vu,i ),i i=1 d v,i u,i +vu,ii = v u+v u. i=1

14 Integration by parts Thus the divergence theorem applied to w = v u gives ( ) v u(s) n(s)ds = (v u) (x)dx Ω Ω = ( v u+v u)(x)dx, which means that v(x) u(x) dx = v(x) u(x)dx Ω Ω v u Ω n ds = a(u,v) v u n ds. Ω UChicago September /65 Ω (8)

15 The variational problem Thus, u can be characterized via u V satisfying a(u,v) = (f,v) L2 (Ω) v V. (9) The companion result, namely a solution to the variational problem in (9) solves Poisson s equation can also be proved [2], under suitable regularity conditions on u. These regularity conditions guarantee the relevant expressions in (1) and (2) are well defined. We will show how this is done in detail in the one-dimensional case. UChicago September /65

16 The variational code The variational formulation translates to code: from dolfin import * # Create mesh and define function space mesh = UnitSquareMesh(32, 32) V = FunctionSpace(mesh, "Lagrange", 1) # Define boundary condition u0 = Constant(0.0) bc = DirichletBC(V, u0, "on_boundary") # Define variational problem u = TrialFunction(V) v = TestFunction(V) you = Expression("(sin( *x[0]))*(sin( *x[1]))") a = inner(grad(u), grad(v))*dx L = (2* * )*you*v*dx # Compute solution u = Function(V) solve(a == L, u, bc) plot(u, interactive=true) UChicago September /65

17 Meaning of dolfin code The code f*dx means Ω f(x)dx where Ω is the domain associated with the function space of which f is a member: f V, Ω = domain(v) In our example, V is defined by V = FunctionSpace(mesh, "Lagrange", 1) and mesh contains the domain information. UChicago September /65

18 Varying variational formulations We have seen that Laplace s equation can be expressed as a variational formulation Find u V such that a(u,v) = F(v) for all v V and furthermore that the variational formulation can be used as a language to describe the problem in code. Now we want to show that this formulation is universal for a broad range of problems. UChicago September /65

19 Formulation of the Pure Neumann Problem: varying V Pure Neumann (or natural) boundary conditions u n = 0 on Ω (10) (i.e., when Γ = Ø) varies the definition of V. In particular, solutions are unique only up to an additive constant, and they can exist only if the right-hand side f in (1) satisfies Ω f(x)dx = = Ω Ω u(x) dx u(x) 1dx UChicago September /65 Ω (11) u n ds = 0.

20 Mean-zero spaces A variational space appropriate for the present case is { } V = v H 1 (Ω) : v(x)dx = 0. (12) For any integrable function g, we define its mean, ḡ, as follows: 1 ḡ := g(x) dx. (13) meas(ω) For any v H 1 (Ω), note that v v V. Then u ū satisfies the variational formulation (9) with V defined as in (12). Ω Ω UChicago September /65

21 Mean-zero spaces Conversely, if u H 2 (Ω) solves (9) then u solves Poisson s equation (1) with a right-hand-side given by with boundary conditions (10). f(x) := f(x) f x Ω (14) Variational form a(, ) coercive on the space V [2]: there is a constant C depending only on Ω and Γ such that v 2 H 1 (Ω) Ca(v,v) v V. (15) Coercivity implies solution is unique: f 0 = 0 = (f,u) L 2 = a(u,u) 1 C u 2 H 1 (Ω). UChicago September /65

22 Coercivity implications In the finite-dimensional case, this uniqueness also implies existence, and a similar result (Lax-Milgram Theorem) holds in the setting of infinite dimensional Hilbert spaces such as V. Moreover, the coercivity condition immediately implies a stability result, namely u H1 (Ω) Ca(u,u) u H1 (Ω) = C (f,u) L 2 u H1 (Ω) C f V, (16) where the dual norm is defined by F V := sup 0 v V F(v) v V. UChicago September /65

23 Continuity condition When coercivity holds, the Lax-Milgram theorem guarantees variational problem (9) has unique solution. Additional continuity condition required: a(u,v) C u H1 (Ω) v H1 (Ω) for all u,v V. (17) Usually this condition is evident, but consider 1 2 u(r 1,r 2 )+(κ(r 1 )+κ(r 2 ))u(r 1,r 2 ) = f(r 1,r 2 ) in Ω = [0, ] [0, ], where κ(r) = r 2 r Here coercivity is easy but continuity requires Hardy s inequality (assuming u = 0 on Ω). UChicago September /65

24 Lax-Milgram theorem Lax-Milgram Theorem Suppose that the variational form a(, ) is coercive (15) and continuous (17) (bounded) on H 1 (Ω). Then the variational problem (9) has a unique solution u for every continuous (bounded) F defined on H 1 (Ω). Moreover, u H1 (Ω) c 1 c 0 sup v H 1 (Ω) F(v) v H1 (Ω), (18) where c 0 is the constant in (15) and c 1 is the constant in (17). The combination of continuity and coercivity correspond to the stability of the numerical scheme. UChicago September /65

25 Advection and Diffusion Many models balance advection and diffusion: ǫ u+β u = f in Ω (19) where β is a vector-valued function. Assume that we have boundary conditions u = g on Γ Ω (Dirichlet) u n = 0 on Ω\Γ (Neumann) (20) The quantity u is being advected in the direction of β. There are in-flow and out-flow parts of the boundary. We need to know how we are allowed to pick Γ. UChicago September /65

26 Variational Formulation of advection-diffusion As before, using the three-step recipe, we define a(u,v) = u(x) v(x)dx Ω ( ) (21) b(u,v) = β(x) u(x) v(x)dx. Ω Alternative formulation: integrate by parts in the advection term. Consider coercivity of the bilinear form a β (u,v) = ǫa(u,v)+b(u,v). Coercivity will guide us regarding choices for Γ. UChicago September /65

27 Coercivity of the Variational Problem Since we already know that a(, ) is coercive on V = { v H 1 (Ω) : v = 0 on Γ }, one approach is to determine conditions under which b(v,v) 0 for all v V. Then a β (v,v) = ǫa(v,v)+b(v,v) ǫa(v,v) cǫ v 2 H. 1 The positivity of b(v,v) will depend on the choice of Γ. UChicago September /65

28 Positivity of the advection form We again invoke the divergence theorem: uvβ nds = (uvβ ) dx Ω Ω = uβ v +vβ u+uv βdx. Ω In particular, b(u,v)+b(v,u) = From (23), we have 2b(v,v) = uvβ nds Ω Ω v 2 β nds (22) UChicago September /65 Ω Ω uv βdx. (23) v 2 βdx. (24)

29 Coercivity of the Variational Problem Define Γ 0 = {x Ω : β(x) n = 0}, Γ ± = {x Ω : ±β(x) n > 0}. An important special case is when β is incompressible, meaning β = 0. (25) We will assume β = 0 from now on. UChicago September /65

30 Incompressible advection In the case β = 0, (24) simplifies to 2b(v,v) = v 2 β nds v 2 β nds, (26) Γ Γ + Γ since, by definition, Thus Γ Γ implies Γ + v 2 β nds 0. b(v,v) 0 for all v V, and thus a β (, ) is coercive on V. We need to control the inflow region of the boundary. UChicago September /65

31 An example Let Ω = [0,1] 2 and β = (1,0). Note that β = 0. Figure 2: Diffusion-advection problem (19) (20) with Γ = Γ Γ + and g(x,y) = y 2( y) and f(x,y) = 1 x. Left: ǫ = 0.1, u ǫ computed using piecewise linears on a mesh. Right: ǫ = 0.001, u ǫ computed using piecewise linears on a mesh. UChicago September /65

32 Numerical pollution Spurious oscillations with too few grid points (on the scale of the mesh, at the outflow boundary) Figure 3: Diffusion-advection problem (19) (20) with Γ = Γ Γ + and g(x,y) = y 2( y) and f(x,y) = 1 x. Left: ǫ = 0.01, u ǫ computed using piecewise linears on a mesh. Right: ǫ = 0.01, u ǫ computed using piecewise linears on a mesh. UChicago September /65

33 Wrong boundary conditions What if no boundary condition on in-flow part: Γ = Γ +. Figure 4: Diffusion-advection problem (19) (20) with Γ = Γ + and g(x,y) = y 2( y) and f(x,y) = 1 x. The solution u ǫ was computed using piecewise linears on a mesh. Left: ǫ = 1.0, Right: ǫ = 0.1. UChicago September /65

34 Don t be fooled Looks reasonable at first. However, look at the scale. The solution is now extremely large. And if we continue to reduce ǫ (exercise) the solution size becomes disturbingly large. Picking different orders of polynomials and values for ǫ tend to give random, clearly spurious results. Thus we conclude that the coercivity condition provides good guidance regarding how to proceed. UChicago September /65

35 Stokes equations The Stokes equations for the flow of a viscous, incompressible, Newtonian fluid can be written u+ p = 0 u = 0. (27) in Ω R d, where u denotes fluid velocity and p denotes pressure [3]. Equations supplemented by boundary conditions, e.g., Dirichlet boundary conditions, u = γ on Ω. Compatibility condition on data (divergence theorem): γ nds = 0. (28) Ω UChicago September /65

36 Stokes variational formulation The variational formulation of (27) takes the form: Find u such that u γ V and p Π such that a(u,v)+b(v,p) = 0 v V, b(u,q) = 0 q Π, (29) where e.g. a(, ) = a (, ) and b(, ) are given by a (u,v) := Ω d i,j=1 u i,j v i,j dx, b(v,q) := Ω d i=1 v i,i qdx. Derived by multiplying (27) by v with a dot product, and integrating by parts as usual. Note that the second equation in (27) and (29) are related by multiplying the former by q and integrating, with no integration by parts. UChicago September /65

37 Stokes variational spaces The spaces V and Π are as follows. V = H 1 0(Ω) d Π = subset of L 2 (Ω) having mean zero. Latter constraint fixes ambient pressure. Inhomogeneous boundary data dealt with by writing u = û+γ, û V, and (29) becomes Find û such that û V and p Π such that a(û,v)+b(v,p) = a(γ,v) := F(v) v V, b(û,q) = b(γ,q) := G(q) q Π. (30) From now on, we will drop the drop the hats on u. UChicago September /65

38 Mixed Method Formulation General formulation of discretization of (29) is a(u h,v)+b(v,p h ) = F(v) v V h b(u h,q) = G(q) q Π h, (31) where F V and G Π (the primes indicate dual spaces [2]). It is called a mixed method since the variables v and q are mixed together. For the Stokes problem (29), the natural variational formulation is already a mixed method, whereas it is an optional formulation in other settings. UChicago September /65

39 Continuity and coercivity conditions We assume that the bilinear forms satisfy the standard continuity conditions a(u,v) C a u V v V u,v V b(v,p) C b v V p Π v V, p Π. (32) We also assume appropriate coercivity conditions α v 2 V a(v,v) v Z Z h b(v,p) β p Π sup p Π h. v V h v V (33) Note the special spaces Z and Z h. UChicago September /65

40 The spaces Z and Z h Z and Z h are defined by Z = {v V : b(v,q) = 0 q Π} (34) and Z h = {v V h : b(v,q) = 0 q Π h } (35) respectively. Z is the set of divergence free functions in H0(Ω), 1 and u Z such that a(u,v) = F(v) v Z. Functions in Z h are not in general divergence free, but u h Z h such that a(u h,v) = F(v) v Z h. UChicago September /65

41 Variational formulation in Z and Z h Thus the varational problem for Stokes appears to be standard in the spaces Z and Z h : u h Z h such that a(u h,v) = F(v) v Z h. But there is a variational crime in general: Z h Z. So there are two things of concern: The approximate flow u h will not necessarily be incompressible, and we do not know if Z h will provide good approximation. UChicago September /65

42 Canonical variational form The mixed formulation can be posed in the canonical variational form by writing A((u,p),(v,q)) :=a(u,v)+b(v,p)+b(u,q) F((v,q)) :=G(q)+F(v) (36) for all (v,q) V := V Π. This can be solved by direct methods (Gaussian elimination), but the system is not positive definite. However, other algorithms can be used in special cases, as we discuss subsequently. UChicago September /65

43 Taylor-Hood method The first general spaces used for the Stokes equations (29) were the so-called Taylor-Hood spaces, as follows. Let V k h denote C0 piecewise polynomials of degree k on a triangulation T h of a polygonal domain Ω R d. Let and let V h = Π h = { v ( V k h { q V k 1 h : ) d : v = 0 on Ω } Ω } q(x)dx = 0 (37). (38) UChicago September /65

44 Taylor-Hood method theory It can be proved that (33) holds in both two and three dimensions under very mild restrictions on the mesh [1]. Note that Z h Z in this case. The main drawback of Taylor-Hood is that the divergence-free condition can be substantially violated, leading to a loss of mass conservation. The loss of mass conservation can be avoided if we force the divergence constraint to be satisfied by a penalty method. UChicago September /65

45 Taylor-Hood method limitations Another drawback of Taylor-Hood is that the linear system (36) is bigger and not positive definite. It represents a saddle-point problem. We will see that an iterated penalty method has a symmetric, positive definite linear system that is well conditioned. Thus we consider the question of how to force Z h Z for a general space V h. This can be done by choosing Π h = V h. UChicago September /65

46 Comparison of the matrix sizes A B A ρ B t 0 Iterated Penalty Taylor Hood matrix matrix UChicago September /65

47 Mesh restrictions Under mesh restrictions, convergence of velocity can be proved [8, 7, 6, 4] d k inf-sup mesh restrictions 2 1 NO all crossed triangles 2 2 YES some crossed triangles required 2 3 YES new conditions [Guzman, Scott] 2 4 YES no nearly singular vertices 3 6 YES only one T h known Table 2: Mesh restrictions for exact divergence-free piecewise polynomials; d= dimension of Ω, k= degree of polynomials. UChicago September /65

48 Exact div-free elements Taylor-Hood has no analog for the piecewise linear case Malkus crossed triangles show that the inf-sup condition is not necessary for well-posed mixed method Approximation by Z h based on Powell C 1 piecewise quadratic element. We now turn to an algorithm for solving for the velocity and pressure without dealing explicitly with the pressure space Π h. The algorithm is a general optimization technique called the iterated penalty method. UChicago September /65

49 Iterated Penalty Method Consider a mixed method for Stokes of the form (31): a(u h,v)+b(v,p h ) = F(v) v V h b(u h,q) = G(q) q Π h. Let ρ R and ρ > 0. The iterated penalty method is a(u n,v)+ρ( u n, v) Π = F(v) ( v, (w n +ργ)) Π w n+1 = w n +ρ (u n +γ), The pressure is defined by v V h (39) p h = P Π w h, (40) where w h := w n for terminal value of n. UChicago September /65

50 Convergence properties The convergence properties of (39) follow from [2]. Theorem 0.1 Suppose that the forms (31) satisfy (32) and (33). for V h and Π h = V h. Then the algorithm (39) converges for any 0 < ρ < 2ρ for ρ sufficiently large. For the choice ρ = ρ, (39) converges geometrically with a rate given by ( 1 C a β + C ) 2 / a ρ. αβ The following stopping criterion follows from [2]. UChicago September /65

51 Stopping criteria Theorem 0.2 Suppose that the forms (31) satisfy (32) and (33). for V h and Π h = V h. Then the errors in algorithm (39) can be estimated by ( 1 u n u h V β + C ) a u n P Π G Π αβ and p n p h Π ( ) Ca β + C2 a αβ +ρ C b u n P Π G Π. When G(q) = b(γ,q), then P Π G = P Π γ and since u n Π h, u n P Π G Π = P Π (u n +γ) Π (u n +γ) Π. (41) UChicago September /65

52 Convergence and stopping criteria The latter norm in (41) is easier to compute, avoiding the need to compute P Π G. We formalize this observation in the following result. Corollary 0.1 Under the conditions of Theorem (0.2) the errors in algorithm (39) can be estimated by ( 1 u n u h V β + C ) a (u n +γ) Π αβ and p n p h Π ( ) Ca β + C2 a αβ +ρ C b (u n +γ) Π. UChicago September /65

53 Iterated penalty performance UChicago September /65

54 The iterated penalty code mesh = UnitSquareMesh(meshsize, meshsize, "crossed") V = VectorFunctionSpace(mesh, "Lagrange", k) u = TrialFunction(V) v = TestFunction(V) w = Function(V) a = inner(grad(u), grad(v))*dx + r*div(u)*div(v)*dx b = -div(w)*div(v)*dx F = inner(f, v)*dx u = Function(V) pde = LinearVariationalProblem(a, F - b, u, bc) solver = LinearVariationalSolver(pde) # Scott-Vogelius iterated penalty method iters = 0; max_iters = 10; div_u_norm = 1 while iters < max_iters and div_u_norm > 1e-10: # solve and update w solver.solve() w.vector().axpy(-r, u.vector()) # find the Lˆ2 norm of div(u) to check stopping condition div_u_norm = sqrt(assemble(div(u)*div(u)*dx(mesh))) print "norm(div u)=%.2e"%div_u_norm iters += 1 UChicago September /65

55 Test Problem We can use FEniCS to test our algorithm. In our experiments, we try to recover an analytical solution of the Stokes equations [ ] sin(4πx)cos(4πy) u = cos(4πx) sin(4πy) p = πcos(4πx)cos(4πy) by applying u above as a boundary condition and letting [ 28π f = 2 ] sin(4πx)cos(4πy) 36π 2. cos(4πx)sin(4πy) Next is a complete implementation on the unit square with mesh size h = 1 16 and polynomial order k = 6. UChicago September /65

56 Test Problem graphics Unified Stokes solution for the pressure (left) and velocity (right); UChicago September /65

57 Divergence comparison: smooth problem 1 h k Taylor-Hood Scott-Vogelius Unified Stokes e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-11 Table 3: u L 2 (Ω) for varying h and k with 4 iterations of the penalty method for Scott-Vogelius (and therefore also USA) UChicago September /65

58 Divergence comparison: lid problem 1 h k Taylor-Hood n = 1 n = 4 converged n e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-12 6 Table 4: u L 2 (Ω) for Taylor-Hood and u n L 2 (Ω) with n iterations of the penalty method for Scott-Vogelius (and USA) for varying h and k. UChicago September /65

59 Numerical results UChicago September /65

60 The unified Stokes algorithm Scott-Vogelius: very accurate velocity approximation with exact divergence zero, but pressure approximation is discontinuous. Taylor-Hood: pressure approximation is continuous but the velocity does not preserve mass. The unified Stokes algorithm combines the best of these two methods and eliminates the bad features. Velocity approximation is same as for Scott-Vogelius, but pressure is projected onto the continuous pressure space (38) used in the Taylor-Hood method. UChicago September /65

61 Pressure and velocity errors UChicago September /65

62 Unified Stokes Algorithm Scott-Vogelius pressure p h = w for w V h, computed via iterated penalty. Then the unified Stokes pressure ˆp h is defined by (ˆp h,q) L2 (Ω) = ( w,q) L2 (Ω) q Π h. When the inf-sup condition holds independently of the mesh, the unified Stokes pressure ˆp h also satisfies [5] p ˆp h L2 (Ω) C ( ) inf u v H1 (Ω) + inf p q L2 (Ω). β v V h q Π h USA code is one line: pus = project(div(w), FunctionSpace(mesh, "Lagrange", uorder-1) UChicago September /65

63 Linear algebra Iterated penalty solves well-conditioned, symmetric, positive-definite systems. This makes Scott-Vogelius faster than Taylor-Hood. 1/h k Taylor-Hood Scott-Vogelius Unified Stokes e e e e e e e e e+00 Table 5: Runtimes in seconds for varying mesh size h and polynomial degree k (one solve for Taylor-Hood, four iterations for Scott-Vogelius and unified Stokes). Unified Stokes faster due to smaller pressure space. UChicago September /65

64 Conclusions Automated Modeling with FEniCS variational formulation provides language for PDEs also a basis for PDE theory automation allows complicated algorithms to be used sophisticated optimization algorithms give better linear algebra performance merger of NA, CS, OR UChicago September /65

65 References [1] D. Boffi. Three-dimensional finite element methods for the Stokes problem. SIAM J. Num. Anal., 34: , [2] Susanne C. Brenner and L. Ridgway Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, third edition, [3] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Oxford: Pergammon, second edition, [4] D. S. Malkus and E. T. Olsen. Linear crossed triangles for incompressible media. North-Holland Mathematics Studies, 94: , [5] Hannah Morgan and L. Ridgway Scott. Towards the ultimate finite element method for the stokes equations. SISC, submitted, [6] Jinshui Qin. On the convergence of some low order mixed finite elements for incompressible fluids. PhD thesis, Penn State, [7] L. R. Scott and M. Vogelius. Conforming finite element methods for incompressible and nearly incompressible continua. In Large Scale Computations in Fluid Mechanics, B. E. Engquist, et al., eds., volume 22 (Part 2), pages Providence: AMS, [8] L. R. Scott and M. Vogelius. Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. M 2 AN (formerly R.A.I.R.O. Analyse Numérique), 19: , UChicago September /65