A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements

W I S S E N T E C H N I K L E I D E N S C H A F T A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements Matthias Gsell and Olaf Steinbach Institute of Computational Mathematics DD23, International Conference on Domain Decomposition Methods

Outline 1. Motivation 2. Approach Continuous Formulation Discretization Strategies 3. Numerical Example 4. Outlook 2

Outline 1. Motivation 2. Approach Continuous Formulation Discretization Strategies 3. Numerical Example 4. Outlook 3

Motivation See [Berningner, 2008]. To describe the flow of fluid (water) in porous media, we use the Richards equation for the physical pressure p. Richards Equation ( θ(p(x, t)) K (x) ( n(x) t µ kr θ ( p(x, t) )) ( p(x, t) ϱg z) ) = f (x, t) Quantities θ... saturation ϱ... density K... permeability g... gravitational const. n... porosity µ... viscosity k r... relative permeability f... source term Ω Γ 4

Motivation Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p. Heat Equation p(x, t) t ( k ( p(x, t) ) ) p(x, t) = f (x, t) Quantities k... thermal conductivity f... heat source Ω Γ 5

Motivation Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p. Heat Equation p(x, t) t ( k ( p(x, t) ) ) p(x, t) = f (x, t) Quantities k... thermal conductivity f... heat source Ω ( Both equtions have same second order term k ( p(x, t) ) ) p(x, t). Γ 5

Motivation Let Ω R d. For given data f, g N, g D consider the following quasilinear boundary value problem. Model Problem Find p, such that ( k ( p(x) ) ) p(x) = f (x) in Ω p(x) = g D (x) k ( p(x) ) p(x) n ΓN = g N (x) on Γ D on Γ N 6

Motivation Let Ω R d. For given data f, g N, g D consider the following quasilinear boundary value problem. Model Problem Find p, such that ( k ( p(x) ) ) p(x) = f (x) in Ω p(x) = g D (x) k ( p(x) ) p(x) n ΓN = g N (x) on Γ D on Γ N Assumptions on k : R R. ) k L (R), i.e. k(s) C < for almost all s R. ) there exists a constant c > 0 such that 0 < c k(s) for almost all s R. ) k is Lipschitz continuous, i.e. k(s) k(t) L s t. 6

Motivation Consider the weak formulation of the BVP. Weak Formulation Find p Hg 1 D,Γ (Ω), such that D k(p) p v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω). 7

Motivation Consider the weak formulation of the BVP. Weak Formulation Find p Hg 1 D,Γ (Ω), such that D k(p) p v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω). We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as u(x) := κ ( p(x) ) = p(x) 0 k(s) ds. 7

Motivation Consider the weak formulation of the BVP. Weak Formulation Find p Hg 1 D,Γ (Ω), such that D k(p) p v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω). We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as u(x) := κ ( p(x) ) = p(x) k(s) ds. Therefore we get 0 u(x) = κ ( p(x) ) p(x) = k ( p(x) ) p(x). 7

Motivation Transformed Model Problem Find u Hh 1 D,Γ (Ω), such that D u v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω) with h D := κ(g D ). ) Unique solvability from Lax Milgram Lemma. ) Numerical analysis is well known for this problem. ) Apply inverse Kirchhoff transformation to obtain the physical quantity p. 8

Motivation Transformed Model Problem Find u Hh 1 D,Γ (Ω), such that D u v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω) with h D := κ(g D ). ) Unique solvability from Lax Milgram Lemma. ) Numerical analysis is well known for this problem. ) Apply inverse Kirchhoff transformation to obtain the physical quantity p. We considered nonlinearities of the form k : R R. 8

Motivation Transformed Model Problem Find u Hh 1 D,Γ (Ω), such that D u v dx = f v dx + Ω Ω g N v ds x Γ N is satisfied for all v H 1 0,Γ D (Ω) with h D := κ(g D ). ) Unique solvability from Lax Milgram Lemma. ) Numerical analysis is well known for this problem. ) Apply inverse Kirchhoff transformation to obtain the physical quantity p. We considered nonlinearities of the form k : R R. Try nonlinearities of the form k : R Ω R. 8

Outline 1. Motivation 2. Approach Continuous Formulation Discretization Strategies 3. Numerical Example 4. Outlook 9

Approach Continuous Formulation ( Consider a second order term is of the form k ( p(x), x ) ) p(x). Richards equation Heat equation Ω 1 Γ 2 Γ Ω 2 Ω 1 Ω 2 Γ Γ 1 Different soil parameter. Different permeabilities. Γ 1 Different material types. Different conductivities. 10

Approach Continuous Formulation ( Consider a second order term is of the form k ( p(x), x ) ) p(x). Richards equation Heat equation Ω 1 Γ 2 Γ Ω 2 Ω 1 Ω 2 Γ Γ 1 Γ 1 Different soil parameter. Different material types. Different permeabilities. Different conductivities. ) Apply Kirchhoff transformation. u(x) := κ ( p(x) ) = p(x) 0 k(s, x) ds but u(x) k ( p(x), x ) p(x). 10

Approach Continuous Formulation ( Consider a second order term is of the form k ( p(x), x ) ) p(x). Richards equation Heat equation Ω 1 Γ 2 Γ Ω 2 Ω 1 Ω 2 Γ Γ 1 Γ 1 Different soil parameter. Different material types. Different permeabilities. Different conductivities. ) Apply Kirchhoff transformation. u(x) := κ ( p(x) ) = p(x) 0 k(s, x) ds but u(x) k ( p(x), x ) p(x). new approach to exploit the advantages of the Kirchhoff transformation. 10

Approach Continuous Formulation Let Ω R d. For given data f, g N, g D consider the following quasilinear boundary value problem. Model Problem Find p, such that (k(p(x), x) p(x) ) = f (x) p(x) = g D (x) k(p(x), x) p(x) n ΓN = g N (x) in Ω on Γ D on Γ N Ω 1 Γ D Ω 4 Ω 5 Ω 3 Assumption: ) k ( p(x), x ) = K ( ) X Ωi (x) k i p(x) i=1 ) k i L (R) and 0 < c i k i for c i R, Ω 2 Γ N ) k i is Lipschitz continuous. 11

Approach Continuous Formulation Let Ω R d. For given data f, g N, g D consider the following quasilinear boundary value problem. Model Problem Find p, such that (k(p(x), x) p(x) ) = f (x) p(x) = g D (x) k(p(x), x) p(x) n ΓN = g N (x) in Ω on Γ D on Γ N Nonlinear Variational Formulation Find p H 1 g D,Γ D (Ω), such that ã(p, v) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v H 1 0,Γ D (Ω) The linear form ã(, ) is given by ã(p, v) := Ω k(p) p v dx. 11

Approach Continuous Formulation Use Primal Hybrid Formulation to exploit the structure of the nonlinearity. See [Raviart, 1977]. Γ 2 Introduce X := { } v L 2 (Ω) v i H 1 (Ω i ), i = 1,..., N, Ω 1 Γ Ω 2 and Γ 1 { M 0,ΓN := µ N H 2 1 ( Ω i ) q H 0,ΓN (div, Ω) : i=1 q n i = µ on Ω i, i = 1,..., N }. Ω 1 Ω 2 Γ Γ 1 12

Approach Continuous Formulation The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation Find (p, λ) X M 0,ΓN, such that â(p, v) + b(v, λ) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v X b(p, µ) = g D, µ Ω µ M 0,ΓN 13

Approach Continuous Formulation The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation Find (p, λ) X M 0,ΓN, such that â(p, v) + b(v, λ) = ( f, v ) + ( g N, v ) 0,Ω 0,Γ N b(p, µ) = g D, µ Ω v X µ M 0,ΓN The linear form â(, ) and the bilinear form b(, ) are given by â(p, v) := b(p, µ) := N a i (p i, v i ) = i=1 i=1 N k i (p i ) p i v i dx i=1 Ω i N N b i (p, µ) = γ 0 p Ω i, µ i Ωi The bilinear form b(, ) enforces the test function v and the solution u to be continuous (in a weak sense) on the interface and therefore to be in H 1 (Ω). i=1 13

Approach Continuous Formulation The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation Find (p, λ) X M 0,ΓN, such that â(p, v) + b(v, λ) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v X b(p, µ) = g D, µ Ω µ M 0,ΓN Apply the Kirchhoff transformation in each subdomain separately. u i (x) := κ i ( pi (x) ) = p i (x) k i (s) ds and u i (x) = k i ( pi (x) ) p i (x) p i (x) := κ 1 ( i ui (x) ) 0 13

Approach Continuous Formulation The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation Find (p, λ) X M 0,ΓN, such that â(p, v) + b(v, λ) = ( f, v ) + ( g N, v ) 0,Ω 0,Γ N b(p, µ) = g D, µ Ω v X µ M 0,ΓN Rewrite the linear form â(, ) and the bilinear form b(, ) in terms of u. N N â(p, v) = k i (p i ) p i v i dx = u i v i dx =: a(u, v) i=1 Ω i=1 i Ω i N N b(p, µ) = γ 0 p Ω i, µ i Ωi = γ 0 κ 1 Ω i i (u i ), µ Ωi =: c(u, µ) i=1 i=1 13

Approach Continuous Formulation We end up with the following variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u, λ) X M 0,ΓN, such that a(u, v) + b(v, λ) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v X c(u, µ) = g D, µ Ω µ M 0,ΓN 14

Approach Continuous Formulation We end up with the following variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u, λ) X M 0,ΓN, such that a(u, v) + b(v, λ) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v X c(u, µ) = g D, µ Ω µ M 0,ΓN The above variational problem is equivalent to the variational problem Nonlinear Variational Formulation Find p H 1 g D,Γ D (Ω), such that ã(p, v) = ( f, v ) 0,Ω + ( g N, v ) 0,Γ N v H 1 0,Γ D (Ω) which is uniquely solvable. 14

Approach Discretization Strategies How to discretize the corresponding spaces? See [Wohlmuth, 2001]. refine subdomains For each triangulation T h,i of Ω i define X h,i := Sh,i 1 (T h,i ) = {u C(Ω i ) u T P 1 (T ) } T T h,i and set X h := N i=1 ( ) X h,i H0, Ω 1 i Γ D (Ω i ) 15

Approach Discretization Strategies How to discretize the corresponding spaces? See [Wohlmuth, 2001]. Γ D Γ N ϕ 1,1 ϕ 1,2 Γ 12 Γ C Y h 0 (I 12 h ) ψ 1 ψ 2 ψ 3 ψ 4 Ω 1 M h (2,1) { Define M := Γ 21 Ω 2 Γ N For each interface Γ m define and set Γ N ϕ 2,1 ϕ 2,2 ϕ 2,3 ϕ 2,4 Γ 21 Γ C Y h 0 (I 21 h ) = Wh 0,(2,1) m = (k, l) Γ kl = Ω k Ω l with T h,k finer than T h,l M h,m := S 0 h,m (I h,m ) = {λ L 2 (Γ m) λ E P 0 (E) E I h,m M h := m M M h,m }. } 15

Approach Discretization Strategies Assume ũ h := u h + u h,d with u h X h. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u h, λ h ) X h M h, such that a(u h, v h ) + b(v h, λ h ) = ( f, v h ) c(u h, µ h ) = 0 0,Ω + ( g N, v h ) 0,Γ N a(u h,d, v h ) v h X h µ M h 16

Approach Discretization Strategies Assume ũ h := u h + u h,d with u h X h. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u h, λ h ) X h M h, such that a(u h, v h ) + b(v h, λ h ) = ( f, v h ) c(u h, µ h ) = 0 0,Ω + ( g N, v h ) 0,Γ N a(u h,d, v h ) v h X h In the discrete setting the bilinear form b(, ) can be written as b(u h, µ h ) := N ( uh,i, µ h = ( ) uh,k u h,l, µ h )0, Ωi i=1 = ( ) uh Γm, µ h m M 0,Γm. m M µ M h 0,Γm 16

Approach Discretization Strategies Assume ũ h := u h + u h,d with u h X h. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u h, λ h ) X h M h, such that a(u h, v h ) + b(v h, λ h ) = ( f, v h ) c(u h, µ h ) = 0 0,Ω + ( g N, v h ) The same can be done for the linear form c(, ). c(u h, µ h ) := N i=1 ( κ 1 i (u h,i + u h,d,i ), µ h ) 0,Γ N a(u h,d, v h ) v h X h 0, Ω i µ M h = ( κ 1 k (u h,k + u h,d,k ) κ 1 ) l (u h,l + u h,d,l ), µ h 0,Γm m M = ( κ 1 ) (u h + u h,d ) Γm, µ h. m M 0,Γm 16

Approach Discretization Strategies Assume ũ h := u h + u h,d with u h X h. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation Find (u h, λ h ) X h M h, such that a(u h, v h ) + b(v h, λ h ) = ( f, v h ) c(u h, µ h ) = 0 0,Ω + ( g N, v h ) 0,Γ N a(u h,d, v h ) v h X h µ M h To solve the nonlinear problem, we apply Newton s method and solve a sequence of linear problems. 16

Approach Discretization Strategies Linearized Formulation For a given w h X h find (u h, λ h ) X h M h, such that a(u h, v h ) + b(v h, λ h ) = ( f, v h ) 0,Ω + ( g N, v h ) for all (v h, µ h ) X h M h. c [w h ](u h, µ h ) = c [w h ](w h, µ h ) c(w h, µ h ) 0,Γ N a(u h,d, v h ) The linearized bilinear form c [ ](, ) is given by c [w h ](u h, µ h ) := ( κ 1 ( ) k wh,k + u h,d,k uh,k κ 1 ( ) ) l wh,l + u h,d,l uh,l, µ h m M = ( κ 1 ( ) ) wh + u h,d uh Γm, µ h m M 0,Γm. 0,Γm 17

Approach Discretization Strategies To obtain solvability of the variational problem we have to show that b(v h, µ h ) sup γ b µ h v h X h v h Mh for all µ h M h, X c [w h ](v h, µ h ) sup γ c µ h v h X h v h Mh for all µ h M h, X and a(u h, v h ) sup γ a u h v h KerB v h X for all u h KerC [w h ], X sup a(u h, v h ) > 0 v h KerC [w h ] for all u h KerB, with positive constants γ b, γ c and γ a. See [Nicolaides, 1982]. 18

Outline 1. Motivation 2. Approach Continuous Formulation Discretization Strategies 3. Numerical Example 4. Outlook 19

Numerical Example Γ D Consider the instationary Richards equation. Domain: R Ω 1 Ω 3 R Ω 2 Ω4 Γ N Ω = [0, 1] [0, 2] R 2 Soil types: Ω 1... sand, Ω 2... sandy loam, Ω 3... loam, Ω 4... sand Boundary conditions: κ 1 1 ( 0.5 (1 t)) x Γ D,1, t < 1 h D (x, t) = κ 1 2 ( 0.5 (1 t)) x Γ D,2, t < 1 0.0 x Γ D, t 1 h N (x, t) = 0.0 x Ω \ Γ D R Time discretization parameter: Timestep : τ = 0.02 Timesteps : T = 1500 20

Numerical Example Triangulation of the four domains (zoomed in cutout). VIDEO : Solution 21

Outline 1. Motivation 2. Approach Continuous Formulation Discretization Strategies 3. Numerical Example 4. Outlook 22

Outlook Summary Transform quasiliner PDEs to linear PDEs Apply discretization and linearization methods Numerical example Outlook Numerical analysis of the discrete linearized SPP Convergence analysis Preconditioners 23

Outlook Summary Transform quasiliner PDEs to linear PDEs Apply discretization and linearization methods Numerical example Outlook Numerical analysis of the discrete linearized SPP Convergence analysis Preconditioners Thank you for your attention! 23

[1] H. Berninger. Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation. PhD thesis, Freie Universität Berlin, 2008. [2] R. A. Nicolaides. Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal., 19(2):349 357, 1982. [3] P.-A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp., 31(138):391 413, 1977. [4] Barbara I. Wohlmuth. Discretization methods and iterative solvers based on domain decomposition, volume 17 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2001. 23