Approximation of solutions to equations in static thermoelasticity

We propose a fast method of high order approximations for the solution of the stationary thermoelastic system in R^3 in the unknown displacement vector u and temperature T. The problem of determining T is an independent problem of u and can be obtained by solving a Dirichlet problem for the Laplace equation. When the temperature is known, the displacement u is obtained by solving the Lame' system where the gradient of T is treated as a mass force. By using the basis functions introduced in the theory of approximate approximations, we derive fast and accurate high order formulas for the approximation of u and T. The high accuracy of the method and the convergence order 2, 4, 6 and 8 are confirmed by numerical experiments.