A posteriori error estimates for axisymmetric and nonlinear problems (Q5959277)

Suppose \(x_1\), \(x_2\), \(x_3\) denote the cylidrical coordinates in \(\mathbb{R}^3\). The authors first study the axisymmetric solution of \[ -\sum^3_{i=1} {\partial\over\partial x_i} \Biggl(A_i{\partial U\over\partial x_i}\Biggr)= F, \] in a rotation body with mixed Dirichlet and Neumann boundary conditions in order to establish a posteriori estimates for the primal and dual finite element method. Next, they discuss the nonlinear elliptic problem \[ -\nabla\cdot a(\nabla u,x)+ g(u,x)= f(x)\quad\text{in }\Omega,\quad u(x)= 0\quad\text{on }\partial\Omega, \] and the one-dimensional nonlinear parabolic problem \[ \partial_tu= \partial_x(a(u)\partial_x u)+ f(u),\quad x\in [c,d], \] \[ u(c,t)= u(d,t)= 0,\quad t\in [0,T],\quad u(x,0)= u_0(x),\quad x\in [c,d]. \] The references include 46 titles.