Discontinuous Galerkin Method for interface problem of coupling different order elliptic equations

Transcription

1 Discontinuous Galerkin Method for interface problem of coupling different order elliptic equations Igor Mozolevski, Endre Süli Federal University of Santa Catarina, Brazil Oxford University Computing Laboratory, UK 5th European Finite Element Fair Marseille (France),17 19 May 2007

2 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

3 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

4 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

5 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

6 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

7 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

8 Outline Motivation Model problem Well-posedness of interface problem for elliptic equations of different order IPDGFEM for interface problem A priori estimates of the error Some numerical results Conlusions

9 Motivation Some multiphysics models, important for applications, are described by an interface problem of coupling fourth-order/second-order elliptic equations.

10 Motivation Example 1.Coupling between surface and groundwater flows (Navier-Stokes equations for surface flow and Darcy equation for groundwater flow) 2.Oil flow in a wellbore (Navier-Stokes equations for oil flow in a wellbore and Darcy equation for oil transport in a porous media. Coupling surface and groundwater flows Water: Navier-Stokes Ground: Darcy

11 Motivation Example Plate-membrane coupling: modeling the acoustic characteristics of baffled membranes and the surrounding sound fields (sound insulation performance of building elements). Baffled membrane External load Rigid baffle (plate) Membrane Rigid baffle (plate)

12 Domain decomposition Domain decomposition methods are today largely used to solve numerically interface problems of coupling partial differential equations of different order/type. Through this method the solution of the interface problem is reduced to the successive solution of local subproblems with Dirichlet or Neumann conditions on the interface. E. Miglio, A. Quarteroni, and F. Saleri: Coupling of free surface and groundwater flows. Computers & Fluids, 32:73 83, P. Gervasio: Homogeneous and heterogeneous domain decomposition methods for plate bending problems. Comput. Methods Appl. Mech. Engrg., 194: , 2005.

13 Domain decomposition Domain decomposition methods are today largely used to solve numerically interface problems of coupling partial differential equations of different order/type. Through this method the solution of the interface problem is reduced to the successive solution of local subproblems with Dirichlet or Neumann conditions on the interface. E. Miglio, A. Quarteroni, and F. Saleri: Coupling of free surface and groundwater flows. Computers & Fluids, 32:73 83, P. Gervasio: Homogeneous and heterogeneous domain decomposition methods for plate bending problems. Comput. Methods Appl. Mech. Engrg., 194: , 2005.

14 Domain decomposition Domain decomposition methods are today largely used to solve numerically interface problems of coupling partial differential equations of different order/type. Through this method the solution of the interface problem is reduced to the successive solution of local subproblems with Dirichlet or Neumann conditions on the interface. E. Miglio, A. Quarteroni, and F. Saleri: Coupling of free surface and groundwater flows. Computers & Fluids, 32:73 83, P. Gervasio: Homogeneous and heterogeneous domain decomposition methods for plate bending problems. Comput. Methods Appl. Mech. Engrg., 194: , 2005.

15 Present work Our interest is in the mathematical study of interface problems of coupling fourth-order/second-order elliptic equations and, in particular, in developing effective numerical methods to solve them. We will propose here numerical approximations of interface problems based on Interior Penalty Discontinuos Galerkin Finite Element Method.

16 Present work Our interest is in the mathematical study of interface problems of coupling fourth-order/second-order elliptic equations and, in particular, in developing effective numerical methods to solve them. We will propose here numerical approximations of interface problems based on Interior Penalty Discontinuos Galerkin Finite Element Method.

17 Model problem geometry Suppose that Ω is a bounded, open, convex polyhedral domain in R d, d 2, with boundary Ω which is the union of its open (d 1)-dimensional faces. Let us suppose that Ω is the union of its two open convex polyhedral subdomains Ω = Ω 1 Ω2. n Γ Γ Ω Ω Γ n Let So we have Γ = (Ω 1 Ω2 ) 0, Γ1 = ( Ω 1 \ Γ) 0, Γ2 = ( Ω 2 \ Γ) 0. Ω = (Γ 1 Γ2 ). Let us denote by n the unit outward normal vector to Ω or the unit normal vector to Γ outward to Ω 1.

18 Model problem geometry Suppose that Ω is a bounded, open, convex polyhedral domain in R d, d 2, with boundary Ω which is the union of its open (d 1)-dimensional faces. Let us suppose that Ω is the union of its two open convex polyhedral subdomains Ω = Ω 1 Ω2. n Γ Γ Ω Ω Γ n Let So we have Γ = (Ω 1 Ω2 ) 0, Γ1 = ( Ω 1 \ Γ) 0, Γ2 = ( Ω 2 \ Γ) 0. Ω = (Γ 1 Γ2 ). Let us denote by n the unit outward normal vector to Ω or the unit normal vector to Γ outward to Ω 1.

19 Model problem geometry Suppose that Ω is a bounded, open, convex polyhedral domain in R d, d 2, with boundary Ω which is the union of its open (d 1)-dimensional faces. Let us suppose that Ω is the union of its two open convex polyhedral subdomains Ω = Ω 1 Ω2. n Γ Γ Ω Ω Γ n Let So we have Γ = (Ω 1 Ω2 ) 0, Γ1 = ( Ω 1 \ Γ) 0, Γ2 = ( Ω 2 \ Γ) 0. Ω = (Γ 1 Γ2 ). Let us denote by n the unit outward normal vector to Ω or the unit normal vector to Γ outward to Ω 1.

20 Model problem Problem Find u 1 H 1 (Ω 1 ) and u 2 H 2 (Ω 2 ), such that (T u 1 ) = f 1 in Ω 1, (σ u 2 ) = f 2 in Ω 2, u 1 = g 10 on Γ 1, u 2 = g 20, n u 2 = g 21 on Γ 2, u 2 u 1 = g 1, σ u 2 = g 2, n ( (σ u 2 )) T n u 1 = g 3 on Γ.

21 Well-posedness of interface problem Given an elliptic interface problem, the interface conditions will lead to a well-posed problem, if and only if, they satisfy the so called covering conditions or Lopatinski Shapiro conditions Schechter M. A Generalization of the Problem of Transmission Ann. Scuola Norm. Sup. Pisa, v.14,n.3: , Roitberg, Ja. A. and Sheftel, Z. G. Nonlocal problems for elliptic equations and systems. (Russian) Sibirsk. Mat. Ž., 13: , I. Mozolevskiǐ Coupling of degenerate elliptic equations. I, II. Differ. Equations, 13: , , 1977.

22 Well-posedness of interface problem Given an elliptic interface problem, the interface conditions will lead to a well-posed problem, if and only if, they satisfy the so called covering conditions or Lopatinski Shapiro conditions Schechter M. A Generalization of the Problem of Transmission Ann. Scuola Norm. Sup. Pisa, v.14,n.3: , Roitberg, Ja. A. and Sheftel, Z. G. Nonlocal problems for elliptic equations and systems. (Russian) Sibirsk. Mat. Ž., 13: , I. Mozolevskiǐ Coupling of degenerate elliptic equations. I, II. Differ. Equations, 13: , , 1977.

23 Well-posedness of interface problem Given an elliptic interface problem, the interface conditions will lead to a well-posed problem, if and only if, they satisfy the so called covering conditions or Lopatinski Shapiro conditions Schechter M. A Generalization of the Problem of Transmission Ann. Scuola Norm. Sup. Pisa, v.14,n.3: , Roitberg, Ja. A. and Sheftel, Z. G. Nonlocal problems for elliptic equations and systems. (Russian) Sibirsk. Mat. Ž., 13: , I. Mozolevskiǐ Coupling of degenerate elliptic equations. I, II. Differ. Equations, 13: , , 1977.

24 General interface problem for elliptic equations of different orders Let us consider regularly elliptic operators P i (x; D) of order 2m i in Ω i, i = 1, 2. Problem (Interface problem) Find u = (u 1, u 2 ) defined in Ω 1 Ω 2 such that: P 1 (x; D)u 1 = f 1 in Ω 1, P 2 (x; D)u 2 = f 2 in Ω 2, Bj 2 (x; D)u 2 Bj 1 (x; D)u 1 = g j, j = 1,..., m 1 + m 2 on Γ + respective boundary conditions on Γ 1 and Γ 2 ;

25 PDE symbol For any linear partial differential operator P(x; D) = a α (x)d α, x Ω let us denote its symbol by P(x; ξ) = Example α =m The symbol of the operator α m a α (x)ξ α, x Ω, ξ R d P 1 (x; D) = (T (x) ) is P 1 (x; ξ) = T (x)(ξ ξ2 d ), and the symbol of the operator P 2 (x; D) = (σ(x) ) is P 2 (x; ξ) = σ(x)(ξ ξ2 d )2.

26 Lopatinski Shapiro conditions Let us consider an arbitrary x 0 Γ, let n be the unit normal vector to Γ at x 0 outward to Ω 1 and let ξ be a tangent vector to Γ at x 0. The operators P i (x 0 ; D) being regularly elliptic operators of order 2m i implies that, at any point x 0 Γ, the polynomial P i (x 0 ; ξ + τn) with respect of τ has an equal number m i of complex roots with positive and negative imaginary part for any ξ 0, i = 1,2. x 0 Ω 1 Ω 2 n ξ Γ

27 Lopatinski Shapiro conditions Let us denote by P + 1 (x 0; ξ, τ) (P 2 (x 0; ξ, τ) ) the polynomial whose roots are equal to the roots with positive (negative) imaginary part of the polynomial P 1 (x 0 ; ξ + τn) (P 2 (x 0 ; ξ + τn)). Let us denote by B i j (x 0; ξ) the symbols of the respective interface operators and let us denote by B i j (x 0; ξ, τ) = m i k=1 bjk i k 1 (ξ)τ the residual of division of Bj i(x 0; ξ + τn) by P ± i i = 1, 2; j = 1,.., m 1 + m 2. (x 0 ; ξ, τ),

28 Lopatinski Shapiro conditions Let us denote by P + 1 (x 0; ξ, τ) (P 2 (x 0; ξ, τ) ) the polynomial whose roots are equal to the roots with positive (negative) imaginary part of the polynomial P 1 (x 0 ; ξ + τn) (P 2 (x 0 ; ξ + τn)). Let us denote by B i j (x 0; ξ) the symbols of the respective interface operators and let us denote by B i j (x 0; ξ, τ) = m i k=1 bjk i k 1 (ξ)τ the residual of division of Bj i(x 0; ξ + τn) by P ± i i = 1, 2; j = 1,.., m 1 + m 2. (x 0 ; ξ, τ),

29 Lopatinski Shapiro conditions Definition The system of interface operators B i,j 1 (x; D), i = 1, 2; j = 1,..., m 1 + m 2 cover the pair of elliptic operators P 1 (x; D), P 2 (x; D) at the interface Γ,if d(x 0 ; ξ) = det for any x 0 Γ, ξ 0. b11 1 (ξ) b1 21 (ξ)... b1 m 1 +m 2 1 (ξ) b1m 1 1 (ξ) b2m 1 1 (ξ)... bm 1 1 +m 2 m 1 (ξ) b11 2 (ξ) b2 21 (ξ)... b2 m 1 +m 2 1 (ξ) b1m 2 1 (ξ) b2m 2 1 (ξ)... bm 2 1 +m 2 m 2 (ξ) 0

30 Lopatinski Shapiro conditions Example For our model problem in R 2 we have P 1 (x 0 ; ξ, τ) = T (x 0 )(τ 2 + ξ 2 ), P 2 (x 0 ; ξ, τ) = σ(x 0 )(τ 2 + ξ 2 ) 2 P + 1 (x 0; ξ, τ) = τ iξ; P 2 (x 0; ξ, τ) = (τ + iξ) 2 B 1 1 (x 0; ξ, τ) = 1; B 1 2 (x 0; ξ, τ) = 0; B 1 3 (x 0; ξ, τ) = iξt (x 0 ); Therefore d(x 0 ; ξ) = det B 2 1 (x 0; ξ, τ) = 1 B 2 2 (x 0; ξ, τ) = 2ξ 2 σ(x 0 ) 2iσ(x 0 )ξτ B 2 3 (x 0; ξ, τ) = 2iξ 3 σ(x 0 ) + 2σ(x 0 )ξ 2 τ. 1 0 iξt (x 0 ) 1 2ξ 2 σ(x 0 ) 2iξ 3 σ(x 0 )) 0 2iσ(x 0 )ξ 2σ(x 0 )ξ 2 = 2ξ 2 T (x 0 )σ(x 0 ) 0 for any x 0 Γ, ξ 0.

31 Finite element space Let us consider the finite element space S p (Ω, T h, F) = { v L2 (Ω) : v κ F κ Q pκ ( κ) if F 1 κ (κ) = κ ; κ T h }. Here p = (p κ : κ T h ) is the vector of local polynomial approximation, Q p ( κ c ) = span{ x α : 0 α i p, 1 i d} and {T h } h>0 is a shape-regular family of partitions of Ω formed by affine images F κ of a fixed master-element the unit hypercube κ R d.

32 Method formulation Let us denote by E h the set of all open (d 1)-dimensional faces of all elements κ T h. We shall suppose that Γ, Γ 1, Γ 2 E h, that is we consider matched meshes. Definition (The hp version of the IPDGFEM) find u h S p (Ω, T h, F), such as where B(u h, v) = l(v) v S p (Ω, T h, F). B Ω (u h, v) = B Ω1 Γ1 (u h, v) + B Ω2 Γ2 (u h, v) + B Γ (u h, v), l Ω (v) = l Ω1 Γ1 (v) + l Ω2 Γ2 (v) + l Γ (v).

33 DGFEMs for second-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, Arnold, D. N.: An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM J. Numer. Anal.,19:4, ,1982. Oden, J.T. and Babuška, I. and Baumann, C.E. A discontinuous hp finite element method for diffusion problems,j. Comput. Phys.,146:2, , Houston, P. and Schwab, C. and Süli, E. Discontinuous hp-finite Element Methods for Advection-Diffusion Problems, SIAM Journal of Numerical Analysis.,39:6, , Ern, A. and Guermond, J.-L. Discontinuous Galerkin methods for Friedrichs systems I. General Theory,SIAM J. Numer. Anal.,44:2, , 2006.

34 DGFEMs for second-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, Arnold, D. N.: An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM J. Numer. Anal.,19:4, ,1982. Oden, J.T. and Babuška, I. and Baumann, C.E. A discontinuous hp finite element method for diffusion problems,j. Comput. Phys.,146:2, , Houston, P. and Schwab, C. and Süli, E. Discontinuous hp-finite Element Methods for Advection-Diffusion Problems, SIAM Journal of Numerical Analysis.,39:6, , Ern, A. and Guermond, J.-L. Discontinuous Galerkin methods for Friedrichs systems I. General Theory,SIAM J. Numer. Anal.,44:2, , 2006.

35 DGFEMs for second-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, Arnold, D. N.: An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM J. Numer. Anal.,19:4, ,1982. Oden, J.T. and Babuška, I. and Baumann, C.E. A discontinuous hp finite element method for diffusion problems,j. Comput. Phys.,146:2, , Houston, P. and Schwab, C. and Süli, E. Discontinuous hp-finite Element Methods for Advection-Diffusion Problems, SIAM Journal of Numerical Analysis.,39:6, , Ern, A. and Guermond, J.-L. Discontinuous Galerkin methods for Friedrichs systems I. General Theory,SIAM J. Numer. Anal.,44:2, , 2006.

36 DGFEMs for second-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, Arnold, D. N.: An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM J. Numer. Anal.,19:4, ,1982. Oden, J.T. and Babuška, I. and Baumann, C.E. A discontinuous hp finite element method for diffusion problems,j. Comput. Phys.,146:2, , Houston, P. and Schwab, C. and Süli, E. Discontinuous hp-finite Element Methods for Advection-Diffusion Problems, SIAM Journal of Numerical Analysis.,39:6, , Ern, A. and Guermond, J.-L. Discontinuous Galerkin methods for Friedrichs systems I. General Theory,SIAM J. Numer. Anal.,44:2, , 2006.

37 DGFEMs for second-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, Arnold, D. N.: An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM J. Numer. Anal.,19:4, ,1982. Oden, J.T. and Babuška, I. and Baumann, C.E. A discontinuous hp finite element method for diffusion problems,j. Comput. Phys.,146:2, , Houston, P. and Schwab, C. and Süli, E. Discontinuous hp-finite Element Methods for Advection-Diffusion Problems, SIAM Journal of Numerical Analysis.,39:6, , Ern, A. and Guermond, J.-L. Discontinuous Galerkin methods for Friedrichs systems I. General Theory,SIAM J. Numer. Anal.,44:2, , 2006.

38 Bilinear form in Ω 1 Γ1 B Ω1 Γ1 (u h, v) is the bilinear form corresponding to IPDGFEM for the Laplace operator B Ω1 Γ1 (u, v) = T u v dx + κ Ω 1 κ e Ω 1 Γ1 e Ω 1 Γ1 e e ({T u} [v] + λ{t v} [u]) ds µ e T[v] [u] ds. For any interior(boundary) edge we use standard definitions of the mean value and of the jump of scalar and of normal component of vector functions.

39 DGFEMs for forth-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, G. Engel and K. Garikipati and T. Hughes and M. Larson and L. Mazzei and R. Taylor Continuous/discontinuous finite element approximaions of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams ans plates, and strain gradient elasticity, Comput. Meth. Appl. Mech. Engrg.,191, , Süli, E. and Mozolevski, I. hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Mech. Engrg.,196, , 2007.

40 DGFEMs for forth-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, G. Engel and K. Garikipati and T. Hughes and M. Larson and L. Mazzei and R. Taylor Continuous/discontinuous finite element approximaions of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams ans plates, and strain gradient elasticity, Comput. Meth. Appl. Mech. Engrg.,191, , Süli, E. and Mozolevski, I. hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Mech. Engrg.,196, , 2007.

41 DGFEMs for forth-order elliptic equations Baker, G. Finite Element Methods for Elliptic Equations Using Nonconforming Elements, Math. Comp.,31, 44 59, G. Engel and K. Garikipati and T. Hughes and M. Larson and L. Mazzei and R. Taylor Continuous/discontinuous finite element approximaions of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams ans plates, and strain gradient elasticity, Comput. Meth. Appl. Mech. Engrg.,191, , Süli, E. and Mozolevski, I. hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Mech. Engrg.,196, , 2007.

42 Bilinear form in Ω 2 Γ2 B Ω2 Γ2 (u h, v) is the bilinear form corresponding to IPDGFEM for the bi-laplacian B Ω2 Γ2 (u, v) = σ u v dx κ Ω κ 2 + e Ω 2 Γ2 e ({σ u}[v] + λ{σ v},[u]) ds ({σ u}[ v] + λ{σ v}[ u]) ds e Ω 2 Γ2 e + σ (α[u][v] + β[ u][ v]) ds, e Ω 2 Γ2 e Süli, E. and Mozolevski, I. hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Mech. Engrg.,196, , 2007.

43 Bilinear form in Γ B Γ (u h, v) is the bilinear form corresponding to the interface operators B Γ (u, v) = ({ σ u, T u}[v] + λ{ σ v, T v}[u]) ds e Γ e + σ u 2 n v 2 ds+ δ[u][v] ds. e Γ e e Γ e If S 1 and S 2 are two differential interface operators on Γ we denote. {S 1 v 1, S 2 v 2 } e = 1 2 (S 1v 1 e + S 2 v 2 e ) [S 1 v 1, S 2 v 2 ] e = S 2 v 2 e S 1 v 1 e

44 Parameters Here λ = ±1 is a parameter, whose values are chosen so as to ensure that B(, ) has certain desirable properties (such as symmetry or coercivity). The functions µ, α, β, δ 0 are defined on respective edges of E h, and are known as the discontinuity-penalisation parameters. They depend on the discretisation parameters h and p in a manner that will be specified later.

45 Consistency Theorem The IPDGFEM formulation of the interface problem is consistent in the space H 2 (Ω 1 ) H 4 (Ω 2 ) in the sense that any solution (u 1, u 2 ) to the interface problem, such that (u 1, u 2 ) H 2 (Ω 1 ) H 4 (Ω 2 ), solves IPDGFEM problem as well.

46 Energy norm Let us introduce in S p (Ω, T h, F) the norm v 2 = T v 2 dx+ σ v 2 dx κ Ω 1 κ + e Ω 1 Γ1 + e Ω 2 Γ2 e e κ Ω 2 κ (µ e [v] 2 + 1µe {T v} 2 ) ds ( α e [v] 2 + β e [ v] 2) ds + ( 1 { σ v} ) {σ v} 2 ds e Ω 2 Γ2 e α e β e + ) (δ e [v] 2 + 1δe { σ v, T v} 2 ds e Γ e

47 Continuity and coercivity Theorem Let us suppose that the penalty parameters µ e,α e,β e and δ e are defined as follows { p 6 } e µ e = θ { p 2 } e h e, α e = θ h 3 e, β e = θ { p 2 } e h e, δ e = α e Then for λ = ±1 and for any θ the bilinear form is continuous with respect to the norm. If λ = 1, this form is coercive for any θ > 0. If λ = 1, there exists a positive constant θ, such that this form is coercive for any θ > θ.

48 A priori error estimate Theorem Let us suppose that the exact solution (u 1, u 2 ) H 2 (Ω 1 ) H 4 (Ω 2 ) to the interface problem belongs to H t (Ω, T h ), t = (t K : K T h ), and t K 2, p K 1 for K Ω 1, t K 4, p K 2 for K Ω 2. Then, if θ is chosen as in the theorem above, the solution u h S p (Ω, T h, F) of IPDGFEM, in the symmetric and non-symmetric formulations, satisfies the following error bound: u u h 2 C 1 κ Ω 1 2s h K 2 K p 2t u 2 K 3 H t K (K ) + C 2 K κ Ω 2 2s h K 4 K p 2t K 7 K u 2 H t K (K ) where 1 s K min (p K + 1, t K ) and C i, i = 1, 2, are constants dependent only on the coefficients of respective operator in Ω i, the space dimension d, the shape-regularity constant, the local polynomial degree variation constant and t =max t K. EFEF-5 Igor Mozolevski, Endre Süli Discontinuous Galerkin K K Method h for interface problem

49 2D convergence test Ω = (0, 1) 2, Ω 1 = {(x, y) Ω 0 < x < 0.5} Ω 2 = {(x, y) Ω 0.5 < x < 1} n Γ Γ Ω Ω Γ n Exact solution: { sin(πx) sin(2πy), (x, y) Ω1, u = sin 2 (πx) sin 2 (2πy), (x, y) Ω 2.

50 2D convergence test Ω = (0, 1) 2, Ω 1 = {(x, y) Ω 0 < x < 0.5} Ω 2 = {(x, y) Ω 0.5 < x < 1} n Γ Γ Ω Ω Γ n Exact solution: { sin(πx) sin(2πy), (x, y) Ω1, u = sin 2 (πx) sin 2 (2πy), (x, y) Ω 2.

51 Exact Solution EFEF-5 Igor Mozolevski, Endre Süli Discontinuous Galerkin Method for interface problem

52 Convergence Problem: T = 1, σ = 1 Method: λ = 1, θ = 10 p - polynomial approximation order L - refinement level, N el = 2 L 2 L Method SIPG Norm L 2 H 1 H 2 p L Ω 1 Ω 2 Ω 1 Ω 2 Ω

53 Uniform load Problem: T = 1, σ = 2, Method: λ = 1, θ = 10 EFEF-5 f =1 Igor Mozolevski, Endre Süli Discontinuous Galerkin Method for interface problem

54 Uniform load Problem: T = 1, σ = 0.5, Method: λ = 1, θ = 10 EFEF-5 f =1 Igor Mozolevski, Endre Süli Discontinuous Galerkin Method for interface problem

55 Conclusions We presented an IPDGFEM for the interface problem of coupling fourth-order/second-order elliptic equations. In this approach the discontinuous Galerkin method permits to solve the interface problem directly, without the necessity to consider any iteration procedure. We develop the a priori error estimates in energy norm which are of optimal order in h for the Poisson and biharmonic equation in the respective subdomains in the energy norm, and are suboptimal in p. Numerical experiments in 2D confirm our theoretical results.

56 Conclusions We presented an IPDGFEM for the interface problem of coupling fourth-order/second-order elliptic equations. In this approach the discontinuous Galerkin method permits to solve the interface problem directly, without the necessity to consider any iteration procedure. We develop the a priori error estimates in energy norm which are of optimal order in h for the Poisson and biharmonic equation in the respective subdomains in the energy norm, and are suboptimal in p. Numerical experiments in 2D confirm our theoretical results.

57 Conclusions We presented an IPDGFEM for the interface problem of coupling fourth-order/second-order elliptic equations. In this approach the discontinuous Galerkin method permits to solve the interface problem directly, without the necessity to consider any iteration procedure. We develop the a priori error estimates in energy norm which are of optimal order in h for the Poisson and biharmonic equation in the respective subdomains in the energy norm, and are suboptimal in p. Numerical experiments in 2D confirm our theoretical results.

58 Conclusions We presented an IPDGFEM for the interface problem of coupling fourth-order/second-order elliptic equations. In this approach the discontinuous Galerkin method permits to solve the interface problem directly, without the necessity to consider any iteration procedure. We develop the a priori error estimates in energy norm which are of optimal order in h for the Poisson and biharmonic equation in the respective subdomains in the energy norm, and are suboptimal in p. Numerical experiments in 2D confirm our theoretical results.