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14 JEAN-DAVID BENAMOU, BRITTANY D. FROESE, AND ADAM M. OBERMANsolutions presented in [8]. Method (M2) converges to a slightly different, moreconvex, solution.As no exact solution is available for comparison, following [21], we focus on theminimum value of the solution produced by each method. We also consider theminimum values obtained using the monotone method in [21]; see Table 7. Thesolutions obtained with our methods are lower than the solutions obtained with themonotone, wide-stencil method. The monotone method is known to converge toa supersolution of (MA), so we do expect these values to be higher than the truevalues. This is evident in the numerics as the solution obtained with the monotonemethod becomes lower (and closer to the solutions obtained with our methods) asthe stencil width is increased.Method (M2) performs very slowly in this case. It is also much slower than (M1),which requires about the same number of iterations to converge as it did for thesmooth examples. See Figure 7 and Table 8.(a)fig:sol2(b)fig:corner2Figure 6. Solutions when f = 1 with constant boundary values 1.(a) Surface plot of solution. (b) The methods produces a solutionwith slight negative curvature along the line y = x.fig:ex2table:ex2Nu (N)min(M1), 9-Point 17-Point 33-Point(M2) Stencil Stencil Stencil21 0.2892 0.3115 0.2815 0.283941 0.2734 0.3090 0.2807 0.273261 0.2682 0.3082 0.2803 0.271181 0.2655 0.3078 0.2802 0.2704101 0.2639 0.3076 0.2800 0.2700121 0.2629 0.3075 0.2798 0.2697141 0.2621 0.3074 0.2796 0.2695Table 7. Minimum value of u when f = 1 with constant boundaryvalues 1 on an N ×N grid. We include results from the wide stencilmethods of [21] on nine, seventeen, and thirty-three point stencils.