The multiple-scale polynomial Trefftz method for solving inverse heat conduction problems

• We solve BHCP and IHCP under large noise.
• We propose a polynomial Trefftz method for heat conduction problems.
• We propose a multiple scale method to reduce the condition number of the linear system.
• The multiple scale parameters are fully dependant on the collocation.

The polynomial Trefftz method consists of the polynomial type solutions as bases, providing a cheap boundary-type meshless method to solve the heat conduction equation, since the bases automatically satisfy the governing equation. In order to stably solve the backward heat conduction problem (BHCP), and the inverse heat source problem (IHSP) together with the boundary condition recovery problem by a polynomial Trefftz method, which are both known to be highly ill-posed, we introduce a multiple-scale post-conditioner in the resultant linear system to reduce the condition number. Then the conjugate gradient method (CGM) is used to solve the post-conditioned linear system to determine the unknown expansion coefficients. In the multiple-scale polynomial Trefftz method (MSPTM) the scales are determined a priori by the collocation points on space–time boundary, which can retrieve the missing initial data, the unknown time-dependent heat source as well as the boundary condition rather well. Several numerical examples of the inverse heat conduction problems demonstrate that the MSPTM is effective and accurate, even for those of severely ill-posed inverse problems under very large noises.