Accurate and efficient numerical solutions for elliptic obstacle problems

2.1 State-of-the-art methods for elliptic obstacle problems

This subsection summarizes state-of-the-art methods for elliptic obstacle problems focusing on the primal-dual method incorporating \(L^{1}\) -like penalty term (PDL1P) studied by Zosso et al. [14]. Primal-dual splitting methods have a great deal of attention, particularly in the context of total variation (TV) minimization and \(L^{1}\)-type problems in image processing [15–19].

2.2 Accuracy issues

The solution of obstacle problems must lie on or over the obstacle (\(u\ge \varphi \)), which is also one of requirements for numerical solutions. For FD methods and finite element (FE) methods for the obstacle problem (1.3), for example, this requirement can easily be violated when edges of the free boundary does not match with mesh grids. See Figure 1, where the shaded rectangle indicates the obstacle defined on one-dimensional (1D) interval \([x_{0},x_{5}]\):

$$ \varphi (x)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if }x_{0}\le x< p,\\ 1 &\mbox{if }p\le x\le x_{5}, \end{array}\displaystyle \right . $$

(2.6)

which is not matching with the mesh grids \(\{x_{i}: x_{i}=i\cdot h_{x}, i=0,\ldots,5\}\). The figure shows the true solution u (red solid curve) and a numerical solution \(u_{h}\) (blue dashed curve) of the linear obstacle problem (1.3) in 1D. The numerical solution is clearly underestimated and the magnitude of the error \(|u_{h}-u|\) is maximized at \(x=p\):

$$ \max_{x} \bigl|u_{h}(x)-u(x) \bigr| = \bigl|u_{h}(p\bigr)-u(p) \bigr| = \frac{x_{3}-p}{x_{3}-x_{0}}, $$

(2.7)

which is \(\mathcal{O}(h_{x})\).

The above example has motivated the authors to develop an effective numerical algorithm for elliptic obstacle problems in 2D which detects the neighboring set of the free boundary, determines the subgrid proportions (r’s), and updates the solution for an improved accuracy using subgrid FD schemes. Here the main goal is to try to guarantee \(u(\mathbf{x})\ge \varphi (\mathbf{x})\) for all \(\mathbf{x}\in \varOmega \) (whether x is a grid point or not). Since it is often the case that the free boundary is determined only after solving the problem, the algorithm must be a post-process. Details are presented in Section 4.

3.1 The linear obstacle problem

Now, consider the following Jacobi iteration for simplicity. Given an initialization \(u^{0}\), find \(u^{n}\) iteratively as follows.

Note that Algorithm \(\mathcal{L}_{J}\)produces a solution u of which the function value at a point is a simple average of four neighboring values, satisfying the constraint \(u\ge\varphi\).

One can easily prove that the algebraic system obtained from (3.3) is irreducibly diagonally dominant and symmetric positive definite. Since its off-diagonal entries are all nonpositive, the matrix must be a Stieltjes matrix and therefore an M-matrix [23]. Thus relaxation methods of regular splittings (such as the Jacobi, the Gauss-Seidel (GS), and the successive over-relaxation (SOR) iterations) are all convergent and their limits are the same as û and therefore satisfy (3.5). In this article, variants of Algorithm \(\mathcal{L}_{J}\)for the GS and the SOR would be denoted, respectively, by \(\mathcal{L}_{\mathrm{GS}}\) and \(\mathcal{L}_{\mathrm{SOR}}\) \((\omega)\), where ω is an over-relaxation parameter for the SOR, \(1<\omega<2\). For example, \(\mathcal{L}_{\mathrm{SOR}}\) \((\omega)\) is formulated as

Note that the right side of (3.9)(a) involves updated values wherever available. When \(\omega=1\), Algorithm \(\mathcal{L}_{\mathrm{SOR}}\) \((\omega)\) becomes Algorithm \(\mathcal{L}_{\mathrm{GS}}\); that is, \(\mathcal{L}_{\mathrm{SOR}}\) \((1)=\mathcal{L}_{\mathrm{GS}}\).

3.2 The nonlinear obstacle problem

Given FD schemes for \(\mathcal{M}(u)\) as in (3.18), the nonlinear obstacle problem (3.12) can be solved iteratively by the Jacobi iteration.

The superscript \((n-1)\) on the coefficients \(a_{pq,D}\), \(D=W, E, S, N\), indicate that they are obtained using the last iterate \(u^{n-1}\). Algorithm \(\mathcal{N}_{J}\)produces a solution u of which the function value at a point is a weighted average of four neighboring values, satisfying the constraint \(u\ge\varphi\). One can prove the following corollary, using the same arguments introduced in the proof of Theorem 1.

Variants of Algorithm \(\mathcal{N}_{J}\)for the GS and the SOR can be formulated similarly as for the linear obstacle problem; they would be denoted respectively by \(\mathcal{N}_{\mathrm{GS}}\) and \(\mathcal{N}_{\mathrm{SOR}}\) \((\omega)\). In practice, such symmetric coercive optimization problems, the SOR methods are much more efficient than the Jacobi and Gauss-Seidel methods. We will exploit \(\mathcal{L}_{\mathrm{SOR}}\) \((\omega)\) and \(\mathcal{N}_{\mathrm{SOR}}\) \((\omega)\) for numerical comparisons with state-of-the-art methods, by setting the relaxation parameter ω optimal.

3.3 The optimal relaxation parameter ω̂

4.1 The contact set and the neighboring set

As defined in Section 2.2, an interior grid point is a neighboring point when it is not in the contact set but one of its adjacent grid points is in the contact set. Thus the neighboring points can be found more effectively as follows. Visit each point in the contact set; if any one of its four adjacent points is not in the contact set, then the non-contacting point is a neighboring point. The set of all neighboring points is the neighboring set \(N_{h}\).

4.2 Subgrid determination of the free boundary

Let \(\mathbf{x}_{pq}\) be a neighboring point with two of its adjacent points are contact points (\(C_{h}(p+1,q)=C_{h}(p,q-1)=1\)), as in Figure 2. Then we may assume that the real free boundary passes somewhere between the contact points and the neighboring points. We will suggest an effective strategy for the determination of the free boundary in subgrid level.

4.3 Nonuniform FD schemes on the neighboring set

Thus, the post-processing algorithm of the obstacle SOR (3.9), \(\mathcal{L}_{\mathrm{SOR}}(\omega)\), can be formulated by replacing the two terms in the right side of (3.2) with the right sides of (4.9) and (4.10), and computing \(u_{\mathrm{GS},pq}\) in (3.9.a) correspondingly at all neighboring points.

5.1 Linear obstacle problems

We first consider a non-smooth obstacle \(\varphi_{1}:\varOmega \rightarrow \mathbb{R}\) with \(\varOmega =[0,1]^{2}\), defined by

$$ \varphi_{1}(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 5 & \mbox{if $|x-0.6|+|y-0.6|< 0.04$}, \\ 4.5 & \mbox{if $(x-0.6)^{2}+(y-0.25)^{2}< 0.001$}, \\ 4.5 & \mbox{if $y=0.57$ and $0.075< x< 0.13$}, \\ 0 & \mbox{otherwise}. \end{array}\displaystyle \right . $$

(5.3)

We solve the linear obstacle problem varying resolutions. The tolerance is set \(\varepsilon =10^{-6}\) hereafter except for examples in Section 5.3. Table 1 presents the number of iterations and CPU (the elapsed time, measured in second) for the linear problem of the non-smooth obstacle (5.3). One can see from the table that our suggested method requires less iterations and converges about one order faster in the computation time than the PDL1P, a state-of-the-art method. We have also implemented the primal-dual hybrid gradient (PDHD) algorithm in [15, 18, 19] for obstacle problems. The PDL1P turns out to be a simple adaptation of the PDHD and their performances are about the same, particularly when μ is set large. For the resolution \(64\times64\), Figure 3 depicts the numerical solutions of the PDL1P and the obstacle SOR and their contour lines. For this example, both the PDL1P and the obstacle SOR resulted in almost identical solutions.

\(\boldsymbol{(\varepsilon =10^{-6})}\)	PDL1P		Obstacle SOR
Resolution	Iter	CPU	Iter	CPU
32 × 32	1,284	0.13	84	0.005
64 × 64	1,744	0.71	148	0.03
128 × 128	2,111	3.58	268	0.22
256 × 256	2,097	14.09	525	1.70

For PDL1P [14], set μ = 10⁸, \(r_{1}=0.01\), and \(r_{2}=12.5\).

As the second example, we consider the radially symmetric obstacle \(\varphi_{2}:\varOmega \rightarrow \mathbb{R}\) with \(\varOmega =[-2,2]^{2}\) defined by

$$ \varphi_{2}(r)=\left \{ \textstyle\begin{array}{l@{\quad}l} \sqrt{1-r^{2}} & \mbox{if $r\leq r^{\ast}$}, \\ -1 & \mbox{otherwise}, \end{array}\displaystyle \right . $$

(5.4)

where \(r^{\ast}=0.6979651482233\ldots\) , the solution of

$$ \bigl(r^{\ast} \bigr)^{2} \bigl(1-\log \bigl(r^{\ast}/2 \bigr) \bigr)=1. $$

(5.5)

For the obstacle \(\varphi _{2}\), the analytic solution to the linear obstacle problem can be defined as

$$ u^{\ast}(r)=\left \{ \textstyle\begin{array}{l@{\quad}l} \sqrt{1-r^{2}} & \mbox{if $r\leq r^{\ast}$}, \\ -(r^{\ast})^{2}\ln(r/2)/\sqrt{1-(r^{\ast})^{2}} & \mbox{otherwise}, \end{array}\displaystyle \right . $$

(5.6)

when the boundary condition is set appropriately using \(u^{*}\). See Figure 4, in which we give plots of \(\varphi _{2}\) and the true solution \(u^{*}\).

In Table 2, we compare performances of the PDL1P and the obstacle SOR applied for the linear obstacle problem with (5.4). The PDL1P uses the parameters suggested in [14] (\(\mu=0.1\), \(r_{1}=0.008\), \(r_{2}=15.625\)). As one can see from the table, our suggested method takes about one order less CPU time than the PDL1P for the computation of the numerical solution. In Figure 5, we show the numerical solutions \(u_{h}\) and the errors \(u_{h}-u^{*}\) produced by the PDL1P and the obstacle SOR at the \(64\times64\) resolution. The solutions are almost identical and the errors are nonpositive. This implies that the numerical solutions of the obstacle problem are underestimated.

\(\boldsymbol{(\varepsilon =10^{-6})}\)	PDL1P		Obstacle SOR
Resolution	\(\boldsymbol{L^{\infty}}\) -error	Iter (CPU)	\(\boldsymbol{L^{\infty}}\) -error	Iter (CPU)
32 × 32	8.91⋅10−3	715 (0.09)	8.69⋅10−3	62 (0.02)
64 × 64	3.01⋅10−3	1,340 (0.60)	3.05⋅10−3	122 (0.04)
128 × 128	7.66⋅10−4	1,971 (3.51)	7.64⋅10−4	244 (0.22)
256 × 256	1.86⋅10−4	2,072 (14.85)	1.88⋅10−4	489 (1.63)

The PDL1P uses the parameters suggested in [14] (μ = 0.1, \(r_{1}=0.008\), \(r_{2}=15.625\)).

As a more general obstacle problem, we consider the elastic-plastic torsion problem in [22]. The problem is to find the equilibrium position of the membrane between two obstacles φ, ψ that a force v is acting on:

$$ \min_{u} \int_{\varOmega } |\nabla u|^{2}\, d\mathbf{x}- \int_{\varOmega } uv\, d\mathbf{x}, \quad\text{s.t. } \psi\ge u\ge\varphi \text{ in }\varOmega , u=f \text{ on }\varGamma . $$

(5.7)

Let \(\varOmega =[0,1]^{2}\) and the problem consist of two obstacles \(\varphi_{3}:\varOmega \rightarrow \mathbb{R}\), \(\psi_{3}:\varOmega \rightarrow \mathbb{R}\) and the force \(v:\varOmega \rightarrow \mathbb{R}\) defined by \(\varphi_{3}(x,y)=-\operatorname{dist}(x,\partial \varOmega )\), \(\psi _{3}(x,y)=0.2\) and

$$ v(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 300 & \mbox{if $(x,y)\in S=\{(x,y): |x-y|\leq0.1 \wedge x\leq 0.3\}$,} \\ -70e^{y}g(x) & \mbox{if $x\le1-y$ and $(x,y)\notin S$,} \\ 15e^{y}g(x) & \mbox{if $x> 1-y$ and $(x,y)\notin S$,} \end{array}\displaystyle \right . $$

(5.8)

where

$$ g(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} 6x & \mbox{if }0\le x\le1/6, \\ 2(1-3x) & \mbox{if }1/6< x\le1/3, \\ 6(x-1/3) & \mbox{if }1/3< x\le1/2, \\ 2(1-3(x-1/3)) & \mbox{if }1/2< x\le2/3, \\ 6(x-2/3) & \mbox{if }2/3< x\le5/6, \\ 2(1-3(x-2/3)) & \mbox{if }5/6< x\le1. \end{array}\displaystyle \right . $$

(5.9)

See Figure 6, where the obstacles and the force are depicted.

In Table 3, we present performances of the PDL1P and the obstacle SOR applied for the elastic-plastic torsion problem (5.7). For the PDL1P, we use the parameters suggested in [14] (\(\mu=0.1\), \(r_{1}=0.008\), \(r_{2}=15.625\)). As one can see from the table, our suggested method again resulted in the numerical solution about one order faster than the PDL1P measured in the CPU time. In Figure 7, we illustrate the simulated membranes in the equilibrium satisfying (5.7) and their contact sets at resolution \(64\times64\). In Figures 7(b) and 7(d), the upper and lower contact sets are colored in yellow (brightest in gray scale) and blue (darkest in gray scale), respectively. The results produced by the two methods are apparently the same.

\(\boldsymbol{(\varepsilon =10^{-6})}\)	PDL1P		Obstacle SOR
Resolution	Iter	CPU	Iter	CPU
32 × 32	887	0.13	47	0.02
64 × 64	1,287	0.68	98	0.04
128 × 128	1,609	3.43	193	0.22
256 × 256	1,866	17.27	368	1.58

For the PDL1P, we use the parameters suggested in [14] (μ = 0.1, \(r_{1}=0.008\), \(r_{2}=15.625\)).

5.2 Nonlinear obstacle problems

The obstacle SOR is implemented for nonlinear obstacle problems as described in Section 3.2.

In Table 4, we present experiments for which the obstacle SOR is applied for nonlinear obstacle problems with \(\varphi =\varphi _{i}\), \(i=1,2,3\). From a comparison with linear cases presented in Tables 1, 2, and 3, we can see for each of the obstacles that the obstacle SOR iteration for the nonlinear problem converges in a similar number of iterations as for the linear problem. Only the apparent difference is the CPU time; an iteration of the nonlinear solver is about as six time expensive as that of the linear solver, due to the computation of coefficients as in (3.19). For \(\varphi =\varphi _{1}\), the nonlinear solution is plotted in Figure 8. Compared with the linear solutions in Figure 3, the nonlinear solution shows slightly lower function values, which is expected. As the grid point approaches the obstacles, the solution shows an increasing gradient magnitude. This may enlarge weights for far-away grid values as shown in (3.19), which in return acts as a force to reduce function values. The difference between the linear solution and the nonlinear solution, at the \(64\times64\) resolution, is depicted in Figure 9.

\(\boldsymbol{(\varepsilon =10^{-6})}\)	\(\boldsymbol{\varphi _{1}}\)		\(\boldsymbol{\varphi _{2}}\)		\(\boldsymbol{\varphi _{3}}\)
Resolution	Iter	CPU	Iter	CPU	Iter	CPU
32 × 32	79	0.02	62	0.04	47	0.04
64 × 64	133	0.16	121	0.17	98	0.16
128 × 128	257	1.25	239	1.16	192	1.09
256 × 256	513	10.19	477	9.20	368	8.24

5.3 Post-processing algorithm

In Figure 5, one have seen that the error, the difference between the numerical solution and the analytic solution, shows its highest values near the free boundary. The larger error is due to the result of mismatch between the mesh grid and the obstacle edges. In order to eliminate the error effectively, we apply the subgrid FD schemes in Section 4 as a post-processing (PP) algorithm. For the examples presented in this subsection, the numerical solutions are solved as follows: (a) the problem is solved with \(\varepsilon =10^{-5}\) (pre-processing), (b) the free boundary is estimated with \((k_{0},k_{1})=(10,4)\) and subgrid FD schemes are applied at neighboring grid points as in Section 4, and (c) another round of iterations is applied to satisfy the tolerance \(\varepsilon =10^{-7}\).

Figure 10 shows the numerical solutions to the linear problem associated to (5.10) with and without the post-process, and their errors. The numerical solutions without and with the post-process are obtained iteratively satisfying the tolerance \(\varepsilon =10^{-7}\). Notice that the solution without post-process is underestimated and shows a relatively high error: \(\|u-u_{4,\mathrm{true}}\|_{\infty}=0.066\). The error is reduced to \(\|u_{\mathrm{pp}}-u_{4,\mathrm{true}}\|_{\infty}=8.57\times 10^{-7}\) after the post-process.

Figure 11 includes plots of the error (\(u_{h}-u^{*}\)) at the \(50\times50\) resolution for the linear obstacle problem with \(\varphi =\varphi _{2}\), produced by the PDL1P, the obstacle SOR, and the obstacle SOR+PP. The numerical solutions of the PDL1P and the obstacle SOR are almost identical to each other and clearly underestimated, with the maximum discrepancy occurring around the free boundary due to the misfit between the mesh grid and the free boundary. It can be seen from Figure 11(c) that the post-process can eliminate the misfit error very effectively; the remaining error is the truncation error introduced by the second-order FD schemes.

5.4 Parameter choices

In order to verify effectiveness of the calibration, we implement a line search algorithm to find a relaxation parameter ω̂ that converges in the smallest number of iterations with \(\varepsilon =10^{-6}\), the same tolerance as for the results in Table 3. For \(h=1/64\) and \(h=1/128\), the line search algorithm returned the curves as shown in Figure 12 with

$$ [\widehat{\omega},\mbox{iter}] \approx \left \{ \textstyle\begin{array}{l@{\quad}l} [1.6732, 45 ] & \mbox{when }h=1/32,\\ {[1.8217, 95 ]} & \mbox{when }h=1/64,\\ {[1.9009, 189 ]} & \mbox{when }h=1/128. \end{array}\displaystyle \right . $$

(5.14)

Note that when the calibrated parameters are used, the iteration counts of the obstacle SOR presented in Table 3 are 47, 98, and 193, respectively, for \(h=1/32\), \(h=1/64\), and \(h=1/128\). Thus the calibrated optimal parameters in (5.13) are quite accurate for the optimal convergence.