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A mesh xn+1 at the new time level is first generated using the mesh and the physical solution (xn , un ) at the current time level, and the solution un+1 is then obtained at the new time level. Note that this mesh xn+1 adapts only to the current solution un and thus lags in time. This will not generally cause much trouble if the time step is reasonably small or the solution does not have abrupt changes in time. If the lag of the mesh in time causes a serious problem, several iterations of solving for the mesh and the physical PDE at each new time level can be used (cf.

1. Note that ξ = ξ (x) has a steep gradient near x = 0 where ρ(x) attains its maximum, whereas x = x(ξ ) changes much more smoothly. 3 that for certain situations the formulations involving ξ (x) can in fact have some computational advantages. 2 Optimality of equidistribution The popularity of equidistribution is due largely to its optimality properties. 8 1 Fig. 1. ing interpolated or the solution of a PDE being solved, the mesh- and solutiondependent factor in an error estimate typically has the general form N E(Th ) ≡ (N − 1)s ∑ (h j f j )s+1 .

4 Burgers’ equation with an exact solution 15 difference method. 7 shows a computed solution and the corresponding mesh trajectories obtained with a moving mesh of 61 points. 8. 9 shows that when a moving mesh is used, the error at t = 1 in the H 1 seminorm (see Appendix A) converges at the rate O(N −1 ). In contrast, when a uniform mesh is used, the method does not converge for the range of values of N considered. 17) and using α = 1. The error is smaller with adaptive meshes than with a uniform mesh for the same amount of the CPU time and, to reach the same level of error, more CPU time is required when a uniform mesh is used.

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A new decision method for elementary algebra by Seidenberg A.

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