We consider the least-squares variational kernel-based methods for numerical solution of partial differential equations. Indeed, we focus on least-squares principles to develop meshfree methods to find the numerical solution of a general second order ADN elliptic boundary value problem in domain $\Omega \subset \mathbb{R}^d$ under Dirichlet boundary conditions. Most notably, in these principles it is not assumed that differential operator is self-adjoint or positive definite as it would have to be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system allowing us to circumvent the compatibility conditions arising in standard and mixed-Galerkin methods. In particular, the resulting method does not require certain subspaces satisfying any boundary condition. The trial space for discretization is provided via standard kernels that reproduce $H^\tau(\Omega)$, $\tau>d/2$, as their native spaces. Therefore, the smoothness of the approximation functions can be arbitrary increased without any additional task. The solvability of the scheme is proved and the error estimates are derived for functions in appropriate Sobolev spaces. For the weighted discrete least-squares principles, we show that the optimal rate of convergence in $L^2(\Omega)$ is accessible. Furthermore, for $d \le 3$, the proposed method has optimal rate of convergence in $H^k(\Omega)$ whenever $k \le \tau$. The condition number of the final linear system is approximated in terms of discterization quality. Finally, the results of some computational experiments support the theoretical error bounds.
翻译:我们考虑的是以最小平方块为基础的以最小平方块为基底的最小偏差内核方法。 事实上, 我们注重于最小平方方块原则, 以开发无网格原则, 在域$\ Omega\ subset\ mathbb{R ⁇ d$。 最明显的是, 在这些原则中, 我们无法假定差异操作者是自动连接的或肯定的, 因为它必须在Rayleigh- Ritz 设置中进行。 但是, 新的方案导致一个对称和正直方正的直方位变色系统, 使我们能够绕过标准法和混合加勒金法方法中出现的兼容性条件。 特别是, 由此产生的方法并不要求某些符合任何边界条件的子空间。 离散的试验空间由标准内核网块提供, 复制$(\ Omega) $, $>>> d=xxxxxxx 。 因此, 最接近性功能的平滑度值, 在任何高级内测算方法中, 最优的轨道的精确性机率 。