### Table 1. Smooth solution test problems

"... In PAGE 13: .... The test problems. We test our error estimator with the one dimensional test problems described in Tables 1 and 2. The solutions of the problems displayed in Table1 are smooth and the ones of the problems in Table 2 both have kinks at x = 1=2. Table 1.... In PAGE 17: ... We use the convolution kernel 1 xK4;2(x= x) de ned by K4;2(y) = ? 1 12 (2)(y ? 1) + 7 6 (2)(y) ? 1 12 (2)(y + 1); where (2) is the B-spline obtained by convolving the characteristic function of [?1=2; 1=2] with itself once; see the monograph by Wahlbin [17] and the references therein. That this strategy works can be seen in Table1 0, where we also see that the computational e ectivity indexes remain reasonably small in all three cases. Note... In PAGE 23: ... We take the same range for as in the 1-D, namely, E = 2 ! j ln(1=!) j; N = 4 j ln(1=!) j; where now we take ! = maxf!x; !yg. In Table1 5, we show the results for the smooth solution test problem. We see... ..."

### Table 1. Smooth solution test problems.

1995

"... In PAGE 16: .... The test problems. We test our error estimator with the one-dimensional test problems described in Tables 1 and 2. The solutions of the problems displayed in Table1 are smooth, and the solutions of the problems in Table 2 both have kinks at x =1=2. b.... In PAGE 20: ...3 convolution kernel 1 xK4;2(x= x) de ned by K4;2(y)= 1 12 (2)(y 1) + 7 6 (2)(y) 1 12 (2)(y +1); where (2) is the B-spline obtained by convolving the characteristic function of [ 1=2; 1=2] with itself once; see the monograph by Wahlbin [18] and the references therein. That this strategy works can be seen in Table1 0, where we also see that the computational e ectivity indexes remain reasonably small in all three cases. Note that the DG method converges with order three with or without post-processing.... In PAGE 24: .... ALBERT, B. COCKBURN, D. FRENCH, AND T. PETERSON scheme: vi;j + H vi+1;j vi 1;j 2 x ; vi;j+1 vi;j 1 2 y !x vi+1;j 2vi;j + vi 1;j x2 !y vi;j+1 2vi;j + vi;j 1 y2 = fi;j; where we can take, for example, !x =sup (x;y)2 x 2 H1 @f @x(x; y); @f @y(x; y) ; !y =sup (x;y)2 y 2 H2 @f @x(x; y); @f @y(x; y) ; and Hi(p1;p2)=@H @pi (p1;p2)fori =1; 2. We apply this scheme to the two test problems in Table1 4; note that p =(p1;p2). In our numerical examples, in order to reduce the arti cial viscosity of the scheme, we replace f by the exact solution u in the formulae de ning !x and !y.... In PAGE 24: ... We take the same range for as in the one-dimensional case, namely, E =2! j ln(1=!) j; N =4j ln(1=!) j; where now we take ! =maxf!x;!yg. In Table1 5, we show the results for the smooth solution test problem. We see that the e ectivity index remains small and constant as the grid is re ned.... ..."

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### Table 3. Monotone scheme on smooth solution test problems.

"... In PAGE 13: ... Next, we describe our numerical results. In Table3 , we show our results for the smooth solution test problems.... ..."

### Table 3. Monotone scheme on smooth solution test problems.

1995

"... In PAGE 16: ... Next, we describe our numerical results. In Table3 , we show our results for the smooth solution test problems. We see that in each of the three problems, the monotone scheme converges linearly, as expected, and that the computational... ..."

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### Table 15. Monotone scheme on smooth solution test problem

"... In PAGE 23: ... We take the same range for as in the 1-D, namely, E = 2 ! j ln(1=!) j; N = 4 j ln(1=!) j; where now we take ! = maxf!x; !yg. In Table15 , we show the results for the smooth solution test problem. We see... ..."

### Table 9. DG method with k = 1 on smooth solution test problems.

1995

"... In PAGE 19: ... For these kinds of residuals, the e ectivity index can grow like ( x) 1, as was illustrated in our third analytic example. This is precisely what we ob- serve in Table9 , where results for k = 1 and smooth solutions are shown. Here we sampled at h = fxj+1=4;xj+3=4g.... ..."

Cited by 12

### Table 15. Monotone scheme on smooth solution test problem.

1995

"... In PAGE 24: ... We take the same range for as in the one-dimensional case, namely, E =2! j ln(1=!) j; N =4j ln(1=!) j; where now we take ! =maxf!x;!yg. In Table15 , we show the results for the smooth solution test problem. We see that the e ectivity index remains small and constant as the grid is re ned.... ..."

Cited by 12

### Table 9. DG method with k = 1 on smooth solution test problems.

"... In PAGE 17: ... For these kinds of residuals, the e ectivity index can grow like ( x)?1, as was illustrated in our third analytic example. This is precisely what we observe in Table9 where results for k = 1 and smooth solutions are shown. Here we sampled at h = fxj+1=4; xj+3=4g.... ..."

### Table 2. Non-smooth solution test problems

"... In PAGE 13: .... The test problems. We test our error estimator with the one dimensional test problems described in Tables 1 and 2. The solutions of the problems displayed in Table 1 are smooth and the ones of the problems in Table2 both have kinks at x = 1=2. Table 1.... ..."

### Table 17. The optimal for the monotone scheme on the non- smooth solution test problem.

"... In PAGE 24: ... Tables 16 and 17 show that the a posteriori error estimate works well in the non-smooth case. Note, in particular, in Table17 how the percentage of the grid which fails the paraboloid test is approximately halved with the grid spacing. In Figure 4.... ..."