A greedy sparse meshless method for solving heat conduction problems

In this paper, we introduce a greedy approximation algorithm for solving the transient heat conduction problem. This algorithm can overcome on some of challenges of the full meshless kernel-based methods such as ill-conditioning and computational cost associated with the dense linear systems that arise. In addition, the greedy algorithm allows to control the consistency error by explicit calculation. First, the space derivatives of the heat conduction equation are discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. Then a time-stepping method is employed for the time derivative. Stability of the scheme is also discussed. Finally, numerical results are presented in three test cases. These experiments show that greedy approximation approach is accurate and fast, and yields the better conditioning in contrast with the fully meshless methods.