Discretized Boundary Integral Equations

τ₁ τ3

τ2

x³

x⁴ x²

x¹ x⁵

Figure 4.3: Connectivity table.

We use the connectivity table (see Figure 4.3) to determine the mapping between global and local indices of nodes building the triangular mesh. For example, the point x⁴ is the second vertex of the triangleτ2, i.e., x⁴=x²². Furthermore, forϕ4|_τ2 we have

ϕ4|_τ2(x) =ϕ4|_τ2(R₄(ξ)) = ˆϕ2(ξ) =ξ1 for x∈τ2,ξ∈ˆτ .

As in the case of piecewise constant basis functions we can identify a function T_ϕ(∂Ω)∋g_ϕ=

N



k=1

g_kϕ_k

with a vector [g₁, . . . , g_N]^T ∈ C^N. For the approximation of a smooth enough functiong we can either use an interpolation, i.e.,

g_k=g(x^k).

For a function g ∈L²(∂Ω) it is more appropriate to define the projection P_ϕ: L²(∂Ω)→ Tϕ(∂Ω) as

Pϕg:= arg min

gϕ∈Tϕ(∂Ω)

∥g−gϕ∥_L2(∂Ω).

The coefficients for real-valued functions can be found in the same way as for the piecewise constant functions as the solution to the system of linear equations

N



k=1

g_k⟨ϕ_k, ϕ_j⟩_L2(∂Ω)=⟨g, ϕ_j⟩_L2(∂Ω) for j∈ {1, . . . , N}.

In the case of complex-valued functions we can separately search for the real parts and the imaginary parts of the coefficients.

The space T_ϕ(∂Ω) will be used for the approximation of the Dirichlet data, i.e., the values of the solution on∂Ω.

4.3 Discretized Boundary Integral Equations

In the following sections we describe the discretization of the boundary integral equations derived in Section 3.5. Although the collocation method will be mentioned, we will prefer

54 4 Discretization and Numerical Realization

the well studied Galerkin scheme based on the corresponding variational formulations. To approximate the Dirichlet and Neumann boundary data we use continuous piecewise affine functions and piecewise constant functions, respectively.

4.3.1 Interior Dirichlet Boundary Value Problem

The solution to the interior Dirichlet boundary value problem (3.24) is given by the rep-resentation formula (3.25) with unknown Neumann data g_N := γ^1,intu. To compute the missing values we use the Fredholm integral equation of the first kind

(V_κg_N)(x) = 1

2g_D(x) + (K_κg_D)(x) forx∈∂Ω or equivalently



∂Ω

g_N(y)v_κ(x,y) ds_y= 1

2g_D(x) +



∂Ω

g_D(y)∂v_κ

∂ny

(x,y) ds_y forx∈∂Ω (4.3) with the fundamental solution vκ and the normal derivative

∂v_κ

∂n_y(x,y) = 1 4π

e^{iκ∥x−y∥}

∥x−y∥³(1−iκ∥x−y∥)⟨x−y,n(y)⟩.

The corresponding variational formulation reads

⟨V_κg_N, s⟩_∂Ω =

1

2I+K_κ

 g_D, s



∂Ω

for all s∈H^−1/2(∂Ω) or



∂Ω

s(x)



∂Ω

gN(y)vκ(x,y) dsydsx

= 1 2



∂Ω

s(x)g_D(x) ds_x+



∂Ω

s(x)



∂Ω

g_D(y)∂v_κ

∂n_y(x,y) ds_yds_x. We seek the solution in the approximate form

g_N≈g_N,h :=

E



ℓ=1

g_ℓ^Nψ_ℓ∈T_ψ(∂Ω) (4.4)

which can be identified with a vectorg^N∈C^E. Furthermore, we use theL² projection to approximate the given Dirichlet data

gD≈gD,h:=

N



j=1

g_j^Dϕj ∈Tϕ(∂Ω), (4.5)

which corresponds to a vector g^D ∈ C^N. The solution gN,h is given by the Galerkin For the left-hand side of (4.6) we obtain

E The right-hand side of (4.6) yields

N

56 4 Discretization and Numerical Realization

The Galerkin equations (4.6) can thus be rewritten into the system of linear equations Vκ,hg^N=

1

2Mh+Kκ,h

 g^D.

The approximate solution to (3.24) can be obtained using the discretized representation formula

u(x)≈u_h(x) :=

E



ℓ=1

g_ℓ^N



τℓ

v_κ(x,y) ds_y−

N



j=1

g^D_j



∂Ω

ϕ_j(y)∂v_κ

∂n_y(x,y) ds_y. (4.11) Another approach to the discretization of the boundary integral equations is the col-location method. Because the relation (4.3) is valid for all x ∈ ∂Ω, we can insert the midpointsx^k∗ into (4.3) to obtain the system of linear equations

V_κ,hg^N=

1

2M_h+K_κ,h

 g^D

with

C^E×E ∋V_κ,h[k, ℓ] := 1 4π



τℓ

e^{iκ∥xk∗−y∥}

∥x^k∗−y∥ds_y, R^E×N ∋Mh[k, j] :=ϕj(x^k∗),

C^E×N ∋K_κ,h[k, j] :=



∂Ω

ϕ_j(y) 1 4π

e^{iκ∥xk∗−y∥}

∥x^k∗−y∥³(1−iκ∥x^k∗−y∥)⟨x^k∗−y,n(y)⟩ds_y. Contrary to the Galerkin scheme, the system matrix V_κ,h arising from the collocation method is in general non-symmetric and the stability of the method is still an open problem.

Although the collocation scheme can be derived for other boundary integral equations in the same way, in the following sections we prefer the Galerkin discretization.

4.3.2 Interior Neumann Boundary Value Problem

The solution to the interior Neumann boundary value problem (3.32) is given by the representation formula (3.33) or by its discretized version (4.11). To compute the missing Dirichlet datag_D:=γ^0,intu we use the hypersingular equation

(DκgD)(x) = 1

2gN(x)−(K_κ^∗gN)(x) forx∈∂Ω or

−γ^1,int



∂Ω

g_D(y)∂v_κ

∂ny

(x,y) ds_y = 1

2g_N(x)−



∂Ω

g_N(y)∂v_κ

∂nx

(x,y) ds_y for x∈∂Ω

with

The corresponding variational problem reads

⟨D_κgD, t⟩_∂Ω = Inserting the approximations of the Cauchy data (4.4), (4.5) into (4.12) we obtain the Galerkin equations Using Theorem 3.14, the left-hand side of (4.13) yields

N

For the right-hand side of (4.13) we obtain

E

58 4 Discretization and Numerical Realization

Γ_D

Γ_N

E_D (Dirichlet elements) E_N (Neumann elements) ND (Dirichlet nodes) N_N(Neumann nodes)

Figure 4.4: Triangulation of∂Ω for the mixed boundary value problem.

with matricesM_h andK_κ,hdefined by (4.8) and (4.9), respectively. The Galerkin equations (4.13) can now be represented as

D_κ,hg^D=

1

2M^T_h −K^T_κ,h

 g^N.

Note that in the system of linear equations above the transpositions are not conjugate transpositions and only involve reordering of the original matrices.

4.3.3 Interior Mixed Boundary Value Problem

For the solution to the interior mixed boundary value problem (3.37) we consider the symmetric approach given by the variational formulation

a(s, t, q, r) =F(q, r) for all q∈H^−1/2(ΓD), r ∈H^1/2(ΓN) (4.15) with unknown functions s∈H^−1/2(Γ_D),t∈H^1/2(Γ_N), the bilinear form

a(s, t, q, r) :=⟨V_κs, q⟩_ΓD − ⟨K_κt, q⟩_ΓD+⟨K_κ^∗s, r⟩_ΓN+⟨D_κt, r⟩_ΓN, the right-hand side

F(q, r) :=

1

2I+K_κ



˜ g_D, q



ΓD

− ⟨V_κ˜g_N, q⟩_ΓD+

1

2I−K_κ^∗



˜ g_N, r



ΓN

− ⟨D_κ˜g_D, r⟩_ΓN and some fixed prolongations of the Cauchy datag˜D,g˜N.

Before we proceed, we divide the boundary elements into two groups, E_D denoting the set of all elements with the Dirichlet boundary condition and EN denoting the set of Neumann elements. Similarly, we divide the set of boundary nodes into sets ND and NN

(see Figure 4.4). Note that in the following text we also use the symbols E_D,E_N,N_D and NN to denote the cardinality of the corresponding sets.

Note that for smooth enough functions t, swe have t|_ΓD = 0 and s|_ΓN = 0. Thus, we seek the unknown functions in the approximate forms

t≈t_h := 

i∈NN

t_iϕ_i ∈T_ϕ(Γ_N), s≈s_h := 

j∈ED

s_jψ_j ∈T_ψ(Γ_D) (4.16)

represented by vectors t ∈ C^NN and s ∈ C^ED, respectively. Moreover, we define the

N = 0 with Γ_N⁻ denoting the Neumann part of the boundary without elements adjacent to Γ_D and g˜_N,h|_ΓD = 0. Inserting the approximate representations (4.16), (4.17) into the variational formulation (4.15) we obtain the system of Galerkin equations

a(s_h, t_h, ψ_k, ϕ_ℓ) =F(ψ_k, ϕ_ℓ) for allk∈E_D, ℓ∈N_N. (4.18) For the left-hand side of (4.18) we obtain

⟨V_κs, ψ_k⟩_ΓD = 

Note that the matrices from the previous formulae are submatrices of (4.7), (4.9) and (4.14) with dimensions

V_κ,h ∈C^ED×ED, K_κ,h ∈C^ED×NN, D_κ,h∈C^NN×NN. For the right-hand side of (4.18) we have

⟨˜gD, ψk⟩_ΓD = 

60 4 Discretization and Numerical Realization

with Γ_D⁺ denoting the Dirichlet part of the boundary together with Neumann elements adjacent to Γ_D. Again, the matrices from the above relations are submatrices of (4.8), (4.9), (4.7) and (4.14) with dimensions

M¯h ∈R^ED×ND, K¯κ,h∈C^ED×ND, V¯κ,h∈C^ED×EN,

¯¯

Mh ∈R^EN×NN, K¯¯κ,h∈C^EN×NN, D¯κ,h ∈C^NN×ND. The system of Galerkin equations can now be rewritten as



Note that similarly as for the interior Neumann problem the transpositions do not involve the complex conjugation.

4.3.4 Exterior Dirichlet Boundary Value Problem

In the following three sections we describe the solution to exterior boundary value problems.

Since the discretization techniques are identical to those mentioned above, the discussion will be more succinct.

The approximate solution to the exterior Dirichlet boundary value problem (3.43) is given by the discretized representation formula (3.44), i.e.,

u(x)≈u_h(x) :=− with unknown Neumann data g_N,h. To obtain the missing data we use the variational formulation corresponding to the Fredholm integral equation (3.45), i.e.,

⟨V_κg_N, s⟩_∂Ω = Inserting the approximations of the Cauchy data (4.4), (4.5) into (4.20) we obtain the system of Galerkin equations

E

and the related system of linear equations V_κ,hg^N=

with the matrices given by (4.7), (4.8) and (4.9).

4.3.5 Exterior Neumann Boundary Value Problem

The solution to the exterior Neumann boundary value problem (3.48) is given by the representation formula (3.49) and its discretized variant (4.19). To compute the missing Dirichlet data we use the hypersingular boundary integral equation (3.51) and the related variational problem Substituting the approximate boundary data g_D,h,g_N,hinto (4.21) we obtain the Galerkin equations

62 4 Discretization and Numerical Realization

and the system of linear equations Dκ,hg^D=



−1

2M^T_h −K^T_κ,h

 g^N,

given by the matrices (4.14), (4.8) and (4.9). Again, the transpositions do not involve the complex conjugation.

4.3.6 Exterior Mixed Boundary Value Problem

Lastly, let us consider the exterior mixed boundary value problem (3.53). Similarly as in the case of the interior mixed problem we use the symmetric approach given by the variational formulation

a(s, t, q, r) =F(q, r) for all q∈H^−1/2(Γ_D), r ∈H^1/2(Γ_N) (4.22) with unknown functions s∈H^−1/2(Γ_D),t∈H^1/2(Γ_N), the bilinear form

a(s, t, q, r) :=⟨V_κs, q⟩_ΓD− ⟨K_κt, q⟩_ΓD+⟨K_κ^∗s, r⟩_ΓN+⟨D_κt, r⟩_ΓN the right-hand side

F(q, r) :=



−1

2I+K_κ



˜ g_D, q



Γ_D

− ⟨V_κ˜g_N, q⟩_ΓD+



−1

2I−K_κ^∗



˜ g_N, r



Γ_N

− ⟨D_κ˜g_D, r⟩_ΓN and some fixed prolongations of the Cauchy data˜gD,g˜N. Inserting the discretized functions s_h, t_hfrom (4.16) and the prolongationsg˜_D,g˜_Nfrom (4.17) into (4.22) we obtain the system of Galerkin equations

a(s_h, t_h, ψ_k, ϕ_ℓ) =F(ψ_k, ϕ_ℓ) for all k∈E_D, ℓ∈N_N and the related system of linear equations



V_κ,h −K_κ,h K^T_κ,h D_κ,h

 s t



=

 −V¯_κ,h −¹₂M¯_h+ ¯K_κ,h

−¹₂M¯¯^T_h −K¯¯^T_κ,h −D¯_κ,h

 g^N g^D



with the matrices defined in Section 4.3.3. The transpositions in the above given system do not involve the complex conjugation of the elements.

In document 2011JanZapletal TheBoundaryElementMethodfortheHelmholtzEquationin3DMetodahraničníchprvkůproHelmholtzovurovnicive3D VŠB–TechnicalUniversityofOstravaFacultyofElectricalEngineeringandComputerScienceDepartmentofAppliedMathematics (Stránka 65-74)