Time-dependent Dirichlet Boundary Conditions in Finite Element Discretizations Peter Benner and Jan Heiland November 5, 2015 Seminar Talk at Uni Konstanz
Introduction Motivation A controlled physical processes are controlled via interfaces is modelled via PDEs for the state υ with boundary control u F( υ, υ; u) and approximated through, e.g., finite elements F ( v, v; u). For further treatment like optimal control or model reduction, we want to have a standard state space form of F : v = f (v; u) or v = Av + Bu. Heiland Time-dependent Boundary Conditions in FEM 2/23
Introduction Or a bit more concrete The PDE υ = Aυ, υ = u in the domain, at the boundary is spatially discretized to give an ODE with a constraint M v = Av, v = u at the inner nodes, at the boundary nodes however, elimination like v = v i + v u, where v u = G 1 u fulfills the boundary values, does lead to M v i = Av i + AG 1 u MG 1 u, which is not of the standard form. Heiland Time-dependent Boundary Conditions in FEM 3/23
Introduction Functional Analytical Framework Ω R d, d {2, 3} a bounded and smooth domain with boundary Γ. V := W 1,2 (Ω) and H := L 2 (Ω) the state spaces V the dual space of V with respect to the embedding V H, the trace spaces Q = W 1 2,2 (Γ) and Q = Q := L(Q, R), and the trace operator γ : V Q. Heiland Time-dependent Boundary Conditions in FEM 4/23
Introduction A generic PDE Problem Let T > 0 and consider A: V V. For F L 2 (0, T ; V ), for υ 0 H, and U L 2 (0, T ; Q ), find υ with υ(t) V and υ(t) V, a.e. on (0, T ), so that υ(t) Aυ(t) = F(t), γυ(t) = U(t), holds for almost all t (0, T ), and so that υ(0) = υ 0 in H. We will not discuss time regularity here. For illustration purposes, we consider only the linear case. Heiland Time-dependent Boundary Conditions in FEM 5/23
Introduction Convection-diffusion Example Problem Given a diffusion parameter ν, a convection wind β, and a function g prescribing the boundary conditions, find a function ρ that satisfies ρ(t) + β ρ(t) ν ρ(t) = 0, ρ Γ (t) = g(t). In standard weak formulations, the convection diffusion problem resembles the generic PDE with υ(t) W 1,2 (Ω) and, e.g., A defined via ( ) ( ) β υ(t), φ +ν υ(t), φ dω Aυ(t), φ V,V = for all φ V := W 1,2 (Ω) and with n Ω Γ ν ( υ n (t), φ) dγ, denoting the normal derivative. Heiland Time-dependent Boundary Conditions in FEM 6/23
Introduction Finite Element Discretization Finite element space V = span{ψ i } nv i=1 V Assume that the basis {ψ i } nv i=1 is a nodal basis, i.e. the basis functions are associated with nodes of a mesh and they have local support. We consider the decomposition V = V I V Γ, with dimensions n v = n I + n d where V I = span{ψ i } n I of the inner i=1 is the space spanned by the basis functions and V Γ V is the space of the basis functions {ψ i } nv the boundary i=n I +1 that live on Heiland Time-dependent Boundary Conditions in FEM 7/23
Introduction Finite Element Discretization The splitting of the FEM space V = V I V Γ, is applied for ansatz and test spaces. Accordingly the mass matrix M := [( ψ i, ψ j )H ] i,j=1,n v is split with respect to the test space: M = [ MI and, once more, with respect to the trial space, M I = [ M II M I Γ ], i.e. M II and M I Γ are the part associated with the inner dofs and the part relates to the boundary dofs tested against the inner nodes, respectively. M Γ ], Heiland Time-dependent Boundary Conditions in FEM 8/23
Section Subsection similarly for the system matrix A approximating A is decomposed [ ] AI A = and A I = [ ] A II A I Γ as is a right hand side. A Γ Also, we write the FEM solution as n I v = v I + v Γ = v i ψ i + i=1 n v i=n I +1 and freely assign it with the coefficient vector v = v i ψ i, [ vi v Γ ]. Heiland Time-dependent Boundary Conditions in FEM 9/23
Introduction Finite Element Discretization To assign the boundary values, we simply assign the dofs associated with the corresponding boundaries via Gv = u, where G = [ 0 I ] R nv n I,n v Eventually, the discrete equations read: Problem Find v ( [0, T ] R nv ) so that M I v(t) A I v(t) = f (t), v(0) = α, Gv(t) = u(t), where α is a given initial value and u is the control variable. Heiland Time-dependent Boundary Conditions in FEM 10/23
Possible Approaches The direct approach Note that we cannot simply eliminate the boundary nodes in M I v A I v = 0, Gv = v Γ = u. with the partitioning of M I and A I with respect to the inner and boundary nodes, we obtain [ ] [ ] v MII M I I Γ = A v II v I + A I Γ v Γ Γ and, having inserted v Γ = u, M II v I = A II v I + A I Γ u M I Γ u. Heiland Time-dependent Boundary Conditions in FEM 11/23
Possible Approaches Lifting Define a function that fulfills the boundary values for all time: ] [ṽi (t) ṽ(t) =. u(t) Then the difference to the actual solution ˆv I = v I ṽ I solves M II ˆv I = A II ˆv I + Bu, ˆv I (0) = α I + M 1 II M I Γ u(0), with B = [A II M 1 I M I Γ A I Γ ]. the obtained system is independent choice of the lifting ṽ I some choices of ṽ I are readily extended to nonlinear equations Heiland Time-dependent Boundary Conditions in FEM 12/23
Possible Approaches Multipliers If the constraint is incorporated via a multiplier: one can apply a projection scheme: M v(t) Av(t) G T λ(t) = 0, Gv(t) = u(t), Define P := I M 1 G T S 1 G, where S := GM 1 G T, consider v = Pv + (I P)v =: v i + v g, and find that v g = M 1 G T S 1 u and with B := AM 1 G T S 1. M v i P T Av i = P T Bu, Heiland Time-dependent Boundary Conditions in FEM 13/23
Possible Approaches Ultra Weak Formulations Let Φ = W 2,2 (Ω) W 1,2 0 (Ω) and consider the pure diffusion equation. υ a is called very weak solution if Ω for all φ Φ. ( υ, φ ) dω Ω ν ( υ, φ ) dω = Γ ν ( u, φ ) dγ n The abstract equations indicate that a spatial discretization may lead to a standard system The difficulty however, lies in the definition of matching test functions of high regularity with zero boundary values and suitable ansatz functions. Heiland Time-dependent Boundary Conditions in FEM 14/23
Possible Approaches Ultra Weak Formulations With the nonconforming ansatz spaces V W 1,2 0 (Ω), the ultra weak solution is approximated ( ) v, φ dω + ν ( v, φ ) ( ) ( φ) dω = f, φ dω ν u, dγ n Ω for all φ V. Ω which is doable by in standard FEM packages. Note that the solution is approximated by a functions with a zero trace, i.e. v Γ = 0. Ω Γ Heiland Time-dependent Boundary Conditions in FEM 15/23
Possible Approaches Robin Relaxation The Dirichlet conditions can be approximated by a Robin type condition v α v n + v = g or v n 1 (g v) on Γ, α with a parameter α that is intended to go to zero which are incorporated naturally in the weak formulation Heiland Time-dependent Boundary Conditions in FEM 16/23
Numerical Studies Test Setup For β 1 (x) = 1 10 [ x0 + 1 (x 1 + 1) ] and ν 1 = 0.1, x1 1 1 1 1 Γ 0 Γ 1 x0-1 Γ 3-1 Γ 2 1 find approximations to the scalar function ρ satisfying ρ(t) + β 1 ρ(t) ν 1 ρ(t) = 0, ρ Γ0 (t) = u(t), ρ Γ1 (t) = 0, Γ 2 ρ ν Γ3 (t) = 0, on given discretizations N h and N τ of the spatial domain Ω = [ 1, 1] 2 and of the time interval [0, 0.2]. Heiland Time-dependent Boundary Conditions in FEM 17/23
Numerical Studies Test Setup We consider the following schemes lift lifting of the boundary conditions via split mass matrix proj incorporation of the constraint via Lagrange multiplier and projections ncul nonconforming approximation of ultra week solutions pero relaxation via Robin approximation and for a varying space discretization N h, time discretization N τ, and polynomial degrees cg, and a numerically computed reference solution, we investigate the errors e pcg hn h,τn τ := ρ pcg hn h,τn τ ρ ref measured in a numerical approximation of the L 2 (0, 1; L 2 ([ 1, 1] 2 )) norm. Heiland Time-dependent Boundary Conditions in FEM 18/23
Numerical Studies Estimated Order of Convergence e p1 hnh,τ240 10 4 10 6 e p2 hnh,τ240 10 4 10 6 10 0 10 1 10 2 N h proj lift ncul 10 0 10 1 10 2 Convergence tests for the consistent implementations the dashed lines indicate the slope of a quadratic convergence the dotted lines indicate a convergence of order 2.5. ++ Heiland Time-dependent Boundary Conditions in FEM 19/23 N h
Numerical Studies Penalty versus Accuracy 10 0 e p1 hnh,τ120 10 2 10 6 10 2 10 2 α 10 4 10 6 10 2 10 2 α N h = 6 N h = 12 N h = 24 N h = 48 Errors for pero for different values of the penalization parameter α for exact system solves (right) and approximate system solves (left) Heiland Time-dependent Boundary Conditions in FEM 20/23
Numerical Studies Peformance in GMRES av.#its e p1,tol1e 7 h48,τ120 proj 10.6 1.2 10 5 lift 10.6 1.2 10 5 pero 14.2 7.8 10 6 ncul 10.6 1.1 10 5 Performance of GMRES within the various formulations for N h = 48, N τ = 120, and linear elements (cg = 1) The averaged number of iterations per time-step av.#its and the approximation error in the case that the resulting linear equations are solved using GMRES up to a relative residual of tol = 10 7. The colored cells contain the lowest measured values. Heiland Time-dependent Boundary Conditions in FEM 21/23
Conclusion and an outlook There are numerous approaches that approximate a Dirichlet boundary control problem into a standard state space system Penalization schemes are easy to implement but require a wise choice of the parameter Consistent schemes need but a little extra implementation efforts Future work: Investigate the performance of the schemes in optimal control or model reduction setups Analyse the consistency of the reformulations with the infinite dimensional PDE Heiland Time-dependent Boundary Conditions in FEM 22/23
Conclusion and an outlook There are numerous approaches that approximate a Dirichlet boundary control problem into a standard state space system Penalization schemes are easy to implement but require a wise choice of the parameter Consistent schemes need but a little extra implementation efforts Future work: Investigate the performance of the schemes in optimal control or model reduction setups Analyse the consistency of the reformulations with the infinite dimensional PDE Thank you for your attention. Heiland Time-dependent Boundary Conditions in FEM 22/23
Literature F. Ben Belgacem, H. El Fekih, and J.-P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptotic Anal. 34 (2003), no. 2, 121 136. Peter Benner and Jan Heiland, Time-dependent dirichlet conditions in finite element discretizations, ScienceOpen Research (2015). V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer, Berlin, Germany, 1986. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Springer, Dordrecht, Netherlands, 2009. Heiland Time-dependent Boundary Conditions in FEM 23/23