Abstract:
A new version of the least-squares collocation method (C3-ÌÊÍÊ7) with seventh degree polynomials is proposed and implemented. The method has continuity up to the third derivatives of the piecewise polynomial solution in the sense of least squares. This is achieved by using the values of the solution and its derivatives at the vertices of the grid cells as unknown terms. The C3-ÌÊÍÊ7 is fundamentally different from previous versions of the LSCM in the absence of matching conditions. They explicitly require continuity of the solution and its derivatives at several points on the boundaries between neighboring cells. To solve the problem in a domain with a curved boundary a grid is constructed with rectangular cells. The solution of small irregular cells is continued from neighboring independent ones. Verification of the C3-ÌÊÍÊ7 is carried out by solving two-dimensional boundary value problems for a biharmonic equation in a square and in domain with the curvilinear boundary. The condition numbers of a global matrix and transition matrices from values at nodes to coefficients of polynomial expansion are studied. The advantages of the C3-ÌÊÍÊ7 over previous versions of the LSCM are shown.
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