Computers and Fluids

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Computers and Fluids

journal homepage:www.elsevier.com/locate/compfluid

Imposing mixed Dirichlet–Neumann–Robin boundary conditions in a level-set framework

Ásdís Helgadóttir

^a,∗

, Yen Ting Ng

^b

, Chohong Min

^c

, Frédéric Gibou

^d

aFaculty of Industrial Engineering, Mechanical Engineering and Computer Science Computer Science Department, University of Iceland, Reykjavik 107, Iceland

bComputer Science Department, University of California, Santa Barbara, CA 93106, USA

cMathematics Department and Research Institute for Basic Sciences, KyungHee University, Seoul, Korea 130-701, South Korea

dMechanical Engineering Department, University of California, Santa Barbara, CA 93106, USA

a r t i c l e i n f o

Article history:

Received 17 March 2014 Revised 7 May 2015 Accepted 11 August 2015 Available online 18 August 2015

Keywords:

Finite difference Level set

Mixed boundary conditions

a b s t r a c t

We consider the Poisson equation with mixed Dirichlet, Neumann and Robin boundary conditions on ir- regular domains. We describe a straightforward and efficient approach for imposing the mixed boundary conditions using a hybrid finite-volume/finite-difference approach, leveraging on the work of Gibou et al.

(2002) [14], Ng et al. (2009) [30] and Papac et al. (2010) [33]. We utilize three different level set functions to represent the irregular boundary at which each of the three different boundary conditions must be imposed;

as a consequence, this approach can be applied to moving boundaries. The method is straightforward to im- plement, produces a symmetric positive definite linear system and second-order accurate solutions in the L^∞-norm in two and three spatial dimensions. Numerical examples illustrate the second-order accuracy and the robustness of the method.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The Poisson equation is one of the building blocks in partial- differential-equation based modeling of physical phenomena and has countless applications in fluid dynamics, heat transfer, electrostat- ics, wave phenomena and a range of other important engineering problems. Many different approaches have been proposed for solv- ing the Poisson problem subjected to different boundary conditions.

The main methods used to solve the Poisson equation are finite ele- ment (e.g.[1,4,5,8,9,15,17,18,24,34,42]and the references therein) or finite difference/volume methods (e.g.[2,6,12,14,22,26,27,44–46]and the references therein).

The advantage of a finite element approach is that symmetric positive definite linear systems are always constructed and a pos- teriori error estimates can be used to construct mesh refinement criteria that minimize the overall error. Analysis of finite element schemes and order of accuracy of the methods are also possible us- ing norms induced by the solution’s space. The main drawback of fi- nite element methods in arbitrary geometry is the difficulty associ- ated with the computational complexity of the mesh generation. This comes from the fact that the elements must conform to the irregu- lar domains’ boundary and skewed elements can corrupt the accu- racy of the method. This leads to a significant computational burden,

∗ Corresponding author. Tel.: +3545254917.

E-mail address:asdishe@hi.is(Á. Helgadóttir).

especially in the case where frequent refinement is necessary, as it is the case in free boundary problems.

Various methods have been used to enforce the correct boundary conditions at an irregular interface. The immerse boundary method smears out the interface condition using a

δ

-function formulation, leading to a simple numerical scheme. A drawback is the loss of ac- curacy near the boundary (see[35–38,43]). The immerse interface method (IIM)[25]is second-order accurate method, including near the interface, but it is more difficult to implement, especially in three spatial dimensions. IIM produces sparse but neither symmetric nor positive definite linear systems, which are more costly to solve than symmetric positive definite versions. The immerse interface method seeks to minimize the truncation error of static two dimensional problems and is not a robust second-order accurate method[20,27].

Liu et al. presented a method for discretizing the variable coefficient Poisson equation where the solution and its derivatives may have jumps across the interface in[27]. This discretization is particularly important in applications such as two-phase incompressible flow and flame simulations (see e.g.[11,23,31]). This method is straightforward to implement since only the right-hand-side of the linear system is modified, hence preserving the standard symmetric definite pos- itive (SPD) discretization of the Poisson equation on regular domains.

The solutions are first-order accurate in the L^∞-norm. Second order accurate solutions to the Poisson equation with jumps across inter- faces have for example been developed in[21,32]. In those methods, the stencil of the matrix is however much greater and the matrix

http://dx.doi.org/10.1016/j.compfluid.2015.08.007 0045-7930/© 2015 Elsevier Ltd. All rights reserved.

generally not symmetric. Recently, Guittet et al. introduce the Voronoi interface method (VIM)[16]. This method construct a band Voronoi cells near the interface and applies the idea of the ghost fluid method of Kang et al.[23]on those cells, whose faces are orthogonal to the fluxes. As a consequence only the right-hand-side of the solver is modified and the techniques produces second-order accurate solu- tions and symmetric positive definite linear systems.

Gibou et al. proposed a method for imposing Dirichlet bound- ary conditions instead of jump conditions in [14], with applica- tions to free surface flows and diffusion dominated moving bound- ary problems (see e.g.[10,13,28]). This method is straightforward to implement, produces an SPD linear system and second-order accu- rate solutions in the L^∞-norm. In addition, it has been extended to fourth-order accuracy, albeit non-symmetric, in[12]. Finite volume approaches allow Neumann and Robin boundary conditions to be treated in a straight forward manner leading to the development of hybrid finite volume / level set methods in [19,30,33]. In par- ticular, Papac et al.[33]describes a simple method for the case of Robin boundary conditions that produces second-order accurate so- lutions in the L^∞-norm and a SPD linear system. Purvis and Burkhal- ter[39]and later Ng et al.[30]presented a second-order accurate SPD method for imposing Neumann boundary conditions on irregular do- mains in the context of fluid flows in arbitrary shaped solid objects.

This is an advantage over the more complicated method of Jomaa and Macaskill[22], for which non-symmetric linear systems are ob- tained, although the accuracy of the gradients may drop to first-order.

Bedrossian et al. presented an approach for imposing jump condi- tions in the solution and solution’s gradients on irregular domains in [3]and applied this framework to the case of Dirichlet and Neumann boundary conditions as well so this method can be applied to mixed boundary conditions. The linear systems are SPD and the solutions are second-order accurate but the method is not straightforward to implement. In[7]Coco and Russo et al. present a finite difference ghost-cell multigrid approach for the Poisson equation with mixed Neumann and Dirichlet boundary conditions on arbitrary domains.

There the Neumann boundary condition is always a smooth exten- sion of the Dirichlet boundary conditions and vice versa (i.e. there is never a kink in the irregular interface where the two boundary con- ditions meet even though there may be a kink in the interface). The method is second order accurate so is its gradient. So far, no test are shown in spatial three dimensions. None of the previously mentioned solvers have, therefore, shown that they handle three dimensional examples where mixture of all three types of boundary conditions on the irregular interface where kinks can occur where the boundary conditions meet.

In this paper, we focus on the Poisson problem with mixed Dirichlet–Neumann–Robin boundary conditions. Such boundary con- ditions can be encountered for example in the simulation of free sur- face flows on an arbitrarily shaped topography (Dirichlet–Neumann) or the simulation of heat diffusion under convection cooling on part of the computational domain (Robin-Dirichlet or Robin-Neumann).

We describe an approach for imposing mixed Dirichlet and/or Neumann and/or Robin boundary conditions in a straightforward and robust fashion, based on combining and extending some of our prior work into a unified framework. This method is uncon- ditionally stable, produces a SPD linear system and second-order accurate solutions in the L^∞-norm but its gradient is first order accurate.

2. Equations and numerical method

We considered the Poisson problem on a domain^separated into two disjoint subsets^{− and}^{+such that}=⁻∪⁺, and

is the interface between ^{− and}⁺. We employ three implicit functions,

φ

^D,

φ

^{N and}

φ

^R to describe the different regions where the solution u is computed as well as where the different boundary

conditions are imposed (seeFig. 1). In particular, we are interested in solving the Poisson equation only inside⁻= max

{φ

^D,

φ

^N,

φ

^R

}

< 0.

Dirichlet, Neumann and Robin boundary conditions are applied on =

{φ

^D= 0 ∧

φ

^N< 0 ∧

φ

^R< 0

}

∪

{φ

^N= 0 ∧

φ

^D< 0 ∧

φ

^R<

0

}

∪

{φ

^R= 0 ∧

φ

^D< 0 ∧

φ

^N< 0

}

, respectively. Mathematically, the problem is described as solving for the solution u at a location x satisfying:



^u= F x∈



⁻, u= G on

φ

^{D= 0,}

∂

^u

∂

^{n= K on}

φ

^N= 0,

∂

^u

∂

ⁿ⁺

α

^u= M on

φ

^R= 0,

(1)

where

α

> 0.

We consider a finite volume discretization for imposing the Neu- mann and Robin boundary conditions, as in[30,33,40]: Consider a cell C_{i j}=



i−¹2, i +¹2



×



j−¹2, j +¹2



partially covered by the irregular domain⁻. Taking a finite volume approach, i.e. integrating the left hand side ofEq. (1)over C_ijand evoking the divergence theorem, we obtain:



Ci j∩⁻

∇

^·

∇

^{u d}



=



∂(^Ci j∩⁻)n·

∇

^{u d}



,

where d^{and d}refer to the area and length differentials respec- tively, in two spatial dimensions. Since the boundary

∂ 

C_{i j}∩⁻



has two components, the faces of the grid cell

∂

^Ci j∩⁻and the interface with the irregular external boundary C_ij∩, we consider separately the contribution of the two components:



∂(^Ci j∩⁻)n·

∇

^{u d}



≈



∂^Ci j∩⁻n·

∇

^{u d}



+



Ci j∩^NK d



+



Ci j∩^RM d



−

α

^ui, j



Ci j∩^R d



.

By approximating the boundary integral on the grid faces as the prod- uct of the length and the sampled value at the center, we obtain:



∂(^{Ci j}∩⁻)n·

∇

^{u d}



L_i+¹

2, j

ui+1, j− ui, j



^{x − Li−12, j}

ui, j− ui−1, j



^x + Li, j+¹2

u_{i, j+1}− u_{i, j}



^{y − Li, j−12}

u_{i, j}− u_{i, j−1}



^y

−

α

^ui, j



Ci j∩^R d



+



Ci j∩^NK d



+



Ci j∩^RM d



, where on a face



i−¹2



× [ j −¹2, j +¹2], the length fraction L_i−¹

2, jof the face covered by the irregular domain {x|

φ

^(x)≤ 0} is linearly ap- proximated as:

L_i−¹

2, j=

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩



^y

φ

_i−¹_{2, j−}¹₂

φ

i−¹2, j−¹2−

φ

i−¹2, j+¹2

if

φ

i−¹2, j−¹2 < 0 and

φ

i−¹2, j+¹2 > 0,



^y_φ ^{φi− 12 ,j+ 12}

i− 12 ,j+ 12−φ_{i− 1}

2 ,j− 12

if

φ

i−¹2, j−¹2 > 0 and

φ

i−¹2, j+¹2 < 0,



^{y if}

φ

i−¹2, j−¹2 < 0 and

φ

i−¹2, j+¹2 < 0, 0 if

φ

i−¹2, j−¹2 > 0

and

φ

i−¹2, j+¹2 > 0.

(2)

We, therefore, obtain a linear system for which each row represents the following equation:

L_i+1 2, j

ui+1, j− ui, j



^{x − Li−12, j}

ui, j− ui−1, j



^x

φ^N < 0

Γ^N φ^R< 0

Γ^R Γ^D

Ω Ω

φ^D< 0

(a) A computational domain in 2D. (b) Surface plot (red) and 0-level set (black) ofφ^D.

(c) Surface plot (blue) and 0-level set (black) ofφ^N. (d) Surface plot (green) and 0-level set (black) ofφ^R.

Fig. 1. A two-dimensional computational domain and its representation. The solution is computed in⁻and the Dirichlet, Neumann and Robin boundary conditions are applied on^D,^Nand^R, respectively. The level-set functionsφ^D,φ^Nandφ^Rhave been set arbitrarily to 1 in⁺for visualization purposes; in practice these functions are Lipschitz continuous.

+ L_{i, j+}¹

2

u_{i, j+1}− ui, j



^{y − Li, j−12}

u_{i, j}− ui, j−1



^{y −}

α

^ui, j



Ci j∩^R d



≈



Ci j∩⁻F d



−



Ci j∩^NK d



−



Ci j∩^RM d



. ⁽³⁾

The integrals are found by geometric integration and will be detailed inSection 2.1. In particular, in the case where mixed Neumann and Robin boundary conditions are present, a subcell integration of each of the interfaces is crucial for convergence as detailed inSection 2.1.

We impose Dirichlet boundary conditions by modifying Eq. (3) at grid nodes adjacent to the interface^D=

{φ

^D= 0

}

using the ap- proach introduced by Gibou et al.[14]: Consider a case where the interface defined by

φ

^D= 0 crosses in between grid nodes xiand x_i+1 (seeFig. 2), thenEq. (3)is modified to incorporate the value of G_at the interface, i.e. the expression:

L_i+¹

2, j

u_{i+1, j}− ui, j



^{x − Li−12, j}

u_{i, j}− ui−1, j



^{x ,}

is replaced by

L_i+¹

2, j

G_− u_{i, j}



^x − L_i−¹

2, j

u_{i, j}− u_{i−1, j}



^{x , (4)}

x

i

u

i

Ω

u

i+1

x

i+1

Ω ⁺

x

Γ

G

Γ

u

^G_i+1

Δx

Γ

Fig. 2. Treatment of Dirichlet boundary conditions on irregular domains. The given interface value G_is enforced at the interfaceusing the approach of Gibou et al.[14].

where

G_=Gi+1

|φ

i^D

|

+ Gi

|φ

i^D+1

|

|φ

i^D

|

⁺

|φ

i^D+1

|

^,



^x=



^x

φ

^Di

|φ

i^D

|

⁺

|φ

i^D+1

|

^.

Eq. (3), modified byEq. (4)for grid nodes adjacent to a Dirich- let boundary interface, produces a linear system that enforces mixed Dirichlet, Neumann and Robin boundary conditions at irregular interfaces.

Remarks.

• It is straightforward to implement such a scheme on arbitrary ir- regular domains in two and three spatial dimensions and to see that the corresponding linear systems are symmetric positive def- inite. We are using an incomplete Cholesky preconditioned con- jugate gradient method[41]to solve the linear system.

• In the case where a Dirichlet boundary condition is to be imposed in a computational cell along with a Neumann or a Robin bound- ary condition, we only perform the treatment for the Dirichlet boundary condition, in essence ignoring the Neumann and/or Robin boundary conditions altogether. The method is still clearly second-order accurate. Only in the case where Robin and Neu- mann boundary conditions are to be imposed in a computational cell, do we calculate the portion of the interface with each of them to account for the correct flux.

2.1. Geometric integration

In order to compute the different integrals inEq. (3), we use a modified version of the second-order accurate geometric integration introduced in[29]: Integrations are performed by first splitting cells C_{i, j}(in two spatial dimensions and C_{i, j, k}in three spatial dimensions) into simplices, S (i.e. triangles in two spatial dimensions and tetrahe- drons in three spatial dimensions). If the sets S^{or S}^{are not} simplices they are further split into simplices using a linear interpo- lation of

φ

from the vertices of S as described in[29]. The interface’s length inside a simplex or the area of a simplex in two spatial di- mensions can be easily found using basic formulas. It is also straight- forward to compute the surface of interface inside a simplex or the volume of a simplex in three spatial dimensions. Finally, the total in- tegrals are found by adding each integral over all simplices, i.e.



C_{i, j}f d



= 

S∈T

(

^{Ci, j}

)



Sf d



, and

C_{i, j}⁻ f d



⁻⁼ 

S∈T

(

^{Ci, j}

)



S⁻ f d



^−,

where T(C_{i, j}) represents the triangulation of the current cell and f the function to be integrated, i.e. either K or M.

2.1.1. Subcell integration

The integration method described in the previous section involves a single level set function [29]. When both Neumann and Robin boundary conditions are present, then subcell integration is needed to avoid a drop in accuracy to first-order. The subcell integration method chosen in this paper is described is this section:

The integration procedure over a domain is straightforward since we can simply follow the procedure described above, except that⁻ is described by

φ

= max

(φ

^N,

φ

^R

)

^.

To describe the case of integration over the interface, let’s assume that a cell is cut by both

φ

^Nand

φ

^R, as illustrated inFig. 3. The points P₁and P₂intersecting

φ

^Nwith the cell’s boundary are found, defining a linear approximation of

φ

^Nin that cell. Likewise, we find the points P₃and P₄on the cell’s boundary defining a linear approximation of

φ

^R

in that cell. Then, the intersection point P^∗between these two linear approximations is used to compute the contribution of

C_{i, j}^NK d and

C_{i, j}^RM din the cell. Specifically, we use:



C_{i, j}^NK d



≈K

(

^P1

)

+ K

(

^P∗

)

2 P1P^∗ and



C_{i, j}^RM d



≈M

(

^P4

)

+ M

(

^P∗

)

2 P₄P^∗,

P2

P1 P3

P4

P P2

P1 P3

P4

Fig. 3. The left schematic depicts the original integration scheme of[29]over both^N and^R. In this case, the contribution of both Neumann and Robin boundary conditions are overestimated. The schematic on the right depicts the subcell integration described inSection 2.1.1. In this case^Nand^Rare correctly only integrated to the cross section point P^∗.

Table 1

Maximum error and rate of maximum error for dif- ferent resolution for Ex.3.1.1: Mixed Dirichlet and non-homogenous Neumann boundary conditions on smooth interfaces in two spatial dimensions.

Resolution ||u− uh||∞ Order

32² 1.67 × 10⁻³ —

64² 4.24 × 10⁻⁴ 1.98

128² 1.18 × 10⁻⁴ 1.84

256² 2.89 × 10⁻⁵ 2.03

512² 7.23 × 10⁻⁶ 2.00

1024² 1.80 × 10⁻⁶ 2.01

2048² 4.59 × 10⁻⁷ 1.97

where P₁P^∗and P₄P^∗are, respectively, the lengths of the interval be- tween the points P₁, P^∗and P₄, P^∗.

Remarks.

• Special care is needed in the case where both

φ

^{N and}

φ

^{R cut}

through a cell, but do not intersect. In this case, the contribution of each integral is computed separately.

• In three spatial dimensions, the procedure is similar except that planes instead of lines are used as linear approximations.

3. Numerical experiments

We present numerical evidence that the proposed method is second-order accurate in both two and three spatial dimensions.

3.1. Two spatial dimensions

3.1.1. Mixed Dirichlet and non-homogenous Neumann boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define r =



x²+ y²,

φ

^D= −x + .1,

φ

^N= r − .8,

φ

^R= −1, and the exact solution G=

(

^r− .5

)

³for all x, y∈^{.Fig. 4}depicts the solution and highlights different parts of the interface where Dirichlet and non-homogenous Neumann boundary conditions are enforced.

Table 1demonstrates the second-order accuracy of the method in the L^∞-norm.

3.1.2. Mixed Dirichlet and non-homogenous Robin boundary conditions Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define r =



x²+ y²,

φ

^D= −x + .1,

φ

^N= −1,

φ

^R= r − .75, and the exact solution G = exp

(

^{x· y}

)

for all x, y ∈^{. Fig. 5} depicts the solution and highlights different parts of the interface where Dirichlet and non-homogenous Robin boundary conditions are enforced.Table 2demonstrates the second-order accuracy of the method in the L^∞-norm.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

−1

0

1

−1 0 1

−0.04

−0.02 0 0.02

Fig. 4. Plot of the solution, u, and interfaces for example3.1.1. The left figure shows a top view where the two interfaces are easily detected. The red line represent^Dand the blue circle represent^N. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

−1 0 1

−1 0 1

−0.2

−0.1 0 0.1 0.2

Fig. 5. Plot of the solution, u, and interfaces for example3.1.2. The left figure shows a top view where the two interfaces are easily detected. The red line represent^Dand the green circle represent^R. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.1.2: Mixed Dirichlet and non- homogenous Robin boundary conditions on smooth interfaces in two spatial dimensions.

Resolution ||^{u− u}h||∞ Order

16² 4.87 × 10⁻³ —

32² 1.12 × 10⁻³ 2.11

64² 3.15 × 10⁻⁴ 1.83

128² 8.05 × 10⁻⁵ 1.97

256² 2.14 × 10⁻⁵ 1.91

512² 5.39 × 10⁻⁶ 1.99

1024² 1.38 × 10⁻⁶ 1.97

2048² 3.45 × 10⁻⁷ 2.00

3.1.3. Mixed Dirichlet and non-homogenous Neumann boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define the following for x, y ∈^:

r =



x²+ y²

θ

= tan⁻¹



_y

x

 φ

^D=

⎧ ⎪

⎨

⎪ ⎩

− min

(

^r− 0.1, 0.6 + 0.3 cos

(

⁶

θ)

− r

)

^{if x}≥ 0

− min



r− 0.1, 1.1 −



x+ 1.1√ 0.91



2

+ y²



if x< 0

φ

^N=

⎧ ⎨

⎩

−



1.1 −



x− 1.1√ 0.91



2

+ y²



if x≥ 0

−

(

0.6 + 0.3 cos

(

⁶

θ)

^{− r}

)

^{if x}< 0

φ

^R= −1 G =



r²− .25



3

.

Table 3

Maximum error and rate of maximum error for dif- ferent resolution for Ex.3.1.3: Mixed Dirichlet and non-homogenous Neumann boundary conditions on irregular interfaces in two spatial dimensions.

Resolution ||^{u− u}h||∞ Order

64² 5.09 × 10⁻³ —

128² 1.18 × 10⁻³ 2.11

256² 3.54 × 10⁻⁴ 1.74

512² 9.34 × 10⁻⁵ 1.92

1024² 2.50 × 10⁻⁵ 1.90

Fig. 6depicts the solution and highlights different parts of the inter- face where Dirichlet and non-homogenous Neumann boundary con- ditions are enforced.Table 3demonstrates the second-order accuracy of the method in the L^∞-norm.

3.1.4. Mixed Dirichlet and non-homogenous Robin boundary conditions Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define the following for x, y ∈^:

r =



x²+ y²

θ

= tan⁻¹



_y

x



φ

^D =

⎧ ⎪

⎨

⎪ ⎩

− min

(

^r− 0.1, 0.6 + 0.3 cos

(

⁶

θ)

^{− r}

)

^{if x≥ 0}

− min



r− 0.1, 1.1 −



x+ 1.1√ 0.91



²

+ y²



if x< 0

φ

^N= −1;

φ

^R =

⎧ ⎨

⎩

−



1.1 −



x− 1.1√ 0.91



2

+ y²



if x≥ 0

−

(

0.6 + 0.3 cos

(

⁶

θ)

^{− r}

)

^{if x}< 0 G =



r²− .25



3

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

−1

0

1 −1 0 0 1

0.05 0.1 0.15

Fig. 6. Plot of the solution, u, and interfaces for example3.1.3. The left figure shows a top view where the two interfaces are easily detected. The red curve represents^{Dand the} blue curve represents^N. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

−1 −0.5 0 0.5 1 −1

0 1

0 0.05 0.1 0.15

Fig. 7. Plot of the solution, u, and interfaces for example3.1.4. The left figure shows a top view where the two interfaces are easily detected. The red curve represents^{Dand the} green curve represents^R. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 4

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.1.4: Mixed Dirichlet and non- homogenous Robin boundary conditions on irregular interfaces in two spatial dimensions.

Resolution ||^u− uh||∞ Order

128² 1.24 × 10⁻³ —

256² 2.92 × 10⁻⁴ 2.09

512² 8.78 × 10⁻⁵ 1.73

1024² 1.98 × 10⁻⁵ 2.15

Fig. 7depicts the solution and highlights different parts of the in- terface where Dirichlet and non-homogenous Robin boundary condi- tions are enforced.Table 4demonstrates the second-order accuracy of the method in the L^∞-norm.

3.1.5. Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define the following for x, y ∈^:

r=



(^x− 0.05)²+(^y+ 0.09)², β=

((y+0.09)⁵+5 ·(x−0.05)⁴·(y+0.09)−10 ·(x−0.05)²·(y+0.09)³)

r⁵ ,

φ^D= −1, φ^N=



r− 0.5 −β 3

 , φ^R= x − 0.1,

Table 5

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.1.5: Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions on irregular interfaces in two spatial di- mensions.

Resolution ||u− uh||∞ Order

64² 6.32 × 10⁻³ —

128² 1.63 × 10⁻³ 1.95

256² 4.44 × 10⁻⁴ 1.88

512² 1.09 × 10⁻⁴ 2.03

1024² 3.13 × 10⁻⁵ 1.80

2048² 7.35 × 10⁻⁶ 2.09

G=( x²+ y²2

− .25)³.

Fig. 8depicts the solution and highlights different parts of the inter- face where non-homogenous Neumann and non-homogenous Robin boundary conditions are enforced.Table 5demonstrates the second- order accuracy of the method in the L^∞-norm.

3.1.6. Mixed Dirichlet, non-homogenous Neumann and

non-homogenous Robin boundary conditions on smooth interfaces Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define r =



x²+ y²,

φ

^D= −x − .3,

φ

^N= r − .8,

φ

^R=

−^x+y√₂ + .1, and the exact solution G =



r⁴− .25



3

for all x, y ∈

^{. Fig. 9} depicts the solution and highlights the different parts of the interface where Dirichlet, non-homegeneous Neumann and non-homogeneous Robin boundary conditions are enforced.Table 6 demonstrates the second-order accuracy of the method in the L^∞-norm.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

−1

0

1 −1 0

1 0

0.01 0.02 0.03 0.04

Fig. 8. Plot of the solution, u, and interfaces for example3.1.5. The left figure shows a top view where the two interfaces are easily detected. The blue curve represent^{Nand the} green line represent^R. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

X

10

^-4

2

1

1

-1 -0.5 0.5 -1 -1

0

0

0

Fig. 9. Plot of the solution, u, and interfaces for example3.1.6. The left figure shows a top view where the three interfaces are easily detected. The red line represents^{D, the blue} line represent^Nand the green circle represent^R. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 6

Maximum error and rate of maximum error for dif- ferent resolution for Ex.3.1.6: Mixed Dirichlet, non- homogenous Neumann and non-homogenous Robin boundary conditions on smooth interfaces in two spatial dimensions.

Resolution ||^u− uh||∞ Order

128² 4.14 × 10⁻⁶ —

256² 9.34 × 10⁻⁷ 2.10

512² 2.50 × 10⁻⁷ 1.95

1024² 6.03 × 10⁻⁸ 2.05

2048² 1.74 × 10⁻⁸ 1.79

3.1.7. Mixed Dirichlet, non-homogenous Neumann and

non-homogenous Robin boundary conditions on non-smooth interfaces Consider the Poisson equation on the domain = [−1, 1] × [−1, 1]. We define the following for x, y ∈^:

r=



(^{x− 0.05})²⁺(^{y+ 0.09})^2, β=

((y+0.09)⁵+5 ·(^x− 0.05)⁴·(^y+ 0.09)− 10 ·(^x− 0.05)²·(^y+ 0.09)³)

r⁵ ,

φ^D=x+ y

√2. − .1,

φ^N=



r− 0.5 −β

3



,

φ^R= x − 0.1, G=(

x²+ y²2

− .25)³.

Table 7

Maximum error and rate of maximum error for dif- ferent resolution for Ex.3.1.7: Mixed Dirichlet, non- homogenous Neumann and non-homogenous Robin boundary conditions on non-smooth interfaces in two spatial dimensions.

Resolution ||u− uh||∞ Order

64² 8.10 × 10⁻³ —

128² 2.13 × 10⁻³ 1.92

256² 5.86 × 10⁻⁴ 1.86

512² 1.44 × 10⁻⁴ 2.02

1024² 4.27 × 10⁻⁵ 1.75

2048² 1.04 × 10⁻⁵ 2.04

Fig. 10 depicts the solution and highlights the different parts of the interface where Dirichlet, non-homegeneous Neumann and non- homogeneous Robin boundary conditions are enforced. Table 7 demonstrates the second-order accuracy of the method in the L^∞-norm.

3.2. Three spatial dimensions

3.2.1. Mixed Dirichlet and non-homogenous Neumann boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −x + .1,

φ

^N= r− .8,

φ

^R= −1, and the exact solution G =

(

^r2− .25

)

³for all x, y, z∈

^{.Fig. 11}depicts the different parts of the interface where Dirichlet and non-homogeneous Neumann boundary conditions are enforced.

Table 8shows the second-order accuracy of the method in the L^∞.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5

1 0.03

0.02

-0.01 0

0

0 -1

-1 -0.5 0.5

1 1

Fig. 10. Plot of the solution, u, and interfaces for example3.1.7. The left figure shows a top view where the three interfaces are easily detected. The red line represents^{D, the blue} curves represent^Nand the green line represent^R. The right figure shows the solution inside⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Plot of the two interfaces for example3.2.1. The red plane represents^Dand the blue sphere represents^N. The top part of the blue sphere and the darkened red circle mark the boundary of⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 8

Maximum error and rate of maximum error for dif- ferent resolution for Ex.3.2.1: Mixed Dirichlet and non-homogenous Neumann boundary conditions on smooth interfaces in three spatial dimensions.

Resolution ||^u− uh||∞ Order

16³ 2.55 × 10⁻² —

32³ 5.15 × 10⁻³ 2.31

64³ 1.43 × 10⁻³ 1.85

128³ 4.54 × 10⁻⁴ 1.66

3.2.2. Mixed Dirichlet and non-homogenous Robin boundary conditions Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −x + .1,

φ

^N=

−1,

φ

^R= r − .5, and the exact solution G =

(

^r2− .25

)

³for all x, y, z

∈^{.Fig. 12}depicts the different parts of the interface where Dirich- let and non-homogeneous Robin boundary conditions are enforced.

Table 9shows the second-order accuracy of the method in the L^∞.

Fig. 12. Plot of the two interfaces for example3.2.2. The red plane represents^Dand the green sphere represents^R. The top part of the green sphere and the darkened red circle mark the boundary of⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 9

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.2.2: Mixed Dirichlet and non- homogenous Robin boundary conditions on smooth interfaces in three spatial dimensions.

Resolution ||u− uh||∞ Order

16³ 2.52 × 10⁻⁴ —

32³ 6.83 × 10⁻⁵ 1.88

64³ 1.78 × 10⁻⁵ 1.84

128³ 4.53 × 10⁻⁶ 1.98

3.2.3. Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −1.,

φ

^N= −x + .1,

φ

^R= r − .8, and the exact solution G =

(

^r2− .25

)

³ for all x, y, z ∈ ^{. Fig. 13} depicts the different parts of the interface where non-homegeneous Neumann and non-homogeneous Robin bound- ary conditions are enforced.Table 10shows the second-order accu- racy of the method in the L^∞.

Fig. 13. Plot of the two interfaces for example3.2.3. The blue plane represents^Nand the green sphere represents^R. The top part of the green sphere, and the darkened blue circle mark the boundary of⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 10

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.2.3: Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions on smooth interfaces in three spatial di- mensions.

Resolution ||^u− uh||∞ Order

16³ 3.12 × 10⁻² —

32³ 1.04 × 10⁻² 1.58

64³ 2.97 × 10⁻³ 1.81

128³ 8.39 × 10⁻⁴ 1.83

Table 11

Maximum error and rate of maximum error for differ- ent resolution for Ex.3.2.4: Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions on smooth interfaces in three spatial di- mensions.

Resolution ||^u− uh||∞ Order

16³ 2.71 × 10⁻² —

32³ 9.55 × 10⁻³ 1.51

64³ 2.75 × 10⁻³ 1.80

128³ 7.89 × 10⁻⁴ 1.80

3.2.4. Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −1.,

φ

^N= r − .8,

φ

^R= −x + .1, and the exact solution G =

(

^r2− .25

)

^{3 for all x,}

y, z∈^{.Fig. 14}depicts the different parts of the interface where non-homegeneous Neumann and non-homogeneous Robin bound- ary conditions are enforced.Table 11shows the second-order accu- racy of the method in the L^∞.

3.2.5. Mixed non-homogenous Neumann and non-homogenous Robin boundary conditions

Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −1.,

φ

^N= −x + .1,

φ

^R= r − .5, and the exact solution G =

(

^r2− .25

)

³ for all x, y, z ∈ ^{. Fig. 15} depicts the different parts of the interface where

Fig. 14. Plot of the two interfaces for example3.2.4. The blue sphere represents^N and the green plane represents^R. The top part of the blue sphere and the darkened green circle mark the boundary of⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Plot of the two interfaces for example3.2.5. The blue plane represents^Nand the green sphere represents^R. The top part of the green sphere and the darkened blue circle mark the boundary of⁻. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

non-homegeneous Neumann and non-homogeneous Robin bound- ary conditions are enforced.Table 12shows the second-order accu- racy of the method in the L^∞.

3.2.6. Mixed Dirichlet, non-homogenous Neumann and

non-homogenous Robin boundary conditions on smooth interfaces Consider the Poisson equation on the domain = [−1, 1] × [−1, 1] × [−1, 1]. We define r =



x²+ y²+ z²,

φ

^D= −^x√+y₂ + .1,

φ

^N= r − .8,

φ

^R= −x + .1, and the exact solution G =

(

^r2− .25

)

^4for

all x, y∈^{.Fig. 16}depicts the different parts of the interface where Dirichlet, non-homegeneous Neumann and non-homogeneous Robin boundary conditions are enforced.Table 13shows the second-order accuracy of the method in the L^∞.