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Abstract
This thesis presents a new collocation method using multivariate splines over triangulation or tetrahedralization for solving partial differential equations. The method is applied to the Poisson equation and extended to the second-order elliptic PDE in non-divergence form, with numerical experiments demonstrating its accuracy and efficiency in both 2D and 3D settings compared to existing spline methods. The thesis also explores solving the Dirichlet problem of the 2D and 3D elliptic Monge-Amp\'ere equation and addresses optimal control design for suppressing singularity formation in chemotaxis governed by the parabolic-elliptic Patlak-Keller-Segel system via flow advection, with the spline collocation method employed to solve the optimality conditions and numerical experiments demonstrating effectiveness.