| Elliptic equations widely exist in many physical and chemical problems. One familiar kind of the elliptic equations is Laplace equationΔ(x1,x2) = 0, (x1,x2) ∈Ω,where Δ = (?)2 /(?)x12 + (?)2/(?)x22 is the Laplace operator. The Laplace equation is also called Potential equation which describles the potentials in physics, for example, the electric potential when Ω contains no electric charges, and the magnetic potential for vanishing current density, etc. Another one is Poisson equationΔu(x1,x2) = f(x1,x2), (x1,x2) ∈Ωwhere f(x1,x2) ∈C0(Ω) is the source term, for example, the charge density in the electrical potential. The general elliptic equation isLu(x1,x2) = f(x1,x2). (x1,x2) ∈Ω. where L is the elliptic operator, which has the divergence form and nondivergence formTo determine the solution of the elliptic equationg uniquely one needs a boundary value condition, for example, the Dirichlet conditionand the Neumann conditionThere are many methods to solve the elliptic equations, for example, the finite difference method, the finite element method and the collocation method. The collocation menthod is a numerical method which search for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years. Collocation methods essentially involves determining an approximate solution by a piecewise polynomial by requiring it to satisfy thr differential equation and boundary conditions exactly at certain points. Collocation methods need't compute numerical integral which increases the workload and effects the precision of coefficient matrix. As a result, the collocation method has easier implementation and is higher convergence (?)ute than the finite element method, and it is widely used for solving both engineering and computing mathematics. Spline collocation method using the nodes of Gauss quadrature formula (Gauss points) as collocation points is named orthogonal spline collocation method (OSC), which has better precision and faster convergence rate than the nomal spline collocation methods.There are many restrictive assumpions in the old orthogonal spline collocation methods of elliptic equations. Only the divergence and self-adjoint form of L is considered. The first partial derivatives b1 ux1 and b2ux2, the mixed partial derivativesα12ux1x2 and α21ux2x1 are not considered.The elliptic nonhonogeneous Dirichlet boundary value problems on rectangles are discussed, whose linear, elliptic, nonself-adjoint, partial differential operator is given in nondivergence norm. The first partial derivatives uxi and the mixed partial derivatives ux1x2 are considered. Assume α12=α31 for simplicity. One piecewise Hermite bicubic orthogonal spline collocation scheme is considered for the approximate solution of the elliptic problems, while the nonhomogeneous Dirichlet boundary condition is approximated by means of the piecewise Hermite cubic interpolant. This method has better precision and smaller workload than those who transform the nonhonogeneous boundary value problems into honogeneous ones. Finally, existence, uniqueness and optimal H error bounds are established for the othogonal spline collocation solution.
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