Abstract
The collocation and least residuals (CLR) method combines the methods of collocations (CM) and least residuals. Unlike the CM, in the CLR method an approximate solution of the problem is found from an overdetermined system of linear algebraic equations (SLAE). The solution of this system is sought under the requirement of minimizing a functional involving the residuals of all its equations. On the one hand, this added complication of the numerical algorithm expands the capabilities of the CM for solving boundary value problems with singularities. On the other hand, the CLR method inherits to a considerable extent some convenient features of the CM. In the present paper, the CLR capabilities are illustrated on benchmark problems for 2D and 3D Navier–Stokes equations, the modeling of the laser welding of metal plates of similar and different metals, problems investigating strength of loaded parts made of composite materials, boundary-value problems for hyperbolic equations.
Highlights
EPJ Web of Conferences simplicity of their application
The collocation and least residuals (CLR) method combines the methods of collocations (CM) and least residuals
Unlike the collocation method (CM), in the CLR method an approximate solution of the problem is found from an overdetermined system of linear algebraic equations (SLAE)
Summary
EPJ Web of Conferences simplicity of their application. The approximate solution may be defined in terms of a polynomial of a sufficiently high degree which is uniquely defined over the entire region of the problem solution (the p-variant of the method). As a matter of fact, the error of the SLAE solution, its condition number, and the magnitude of the residual functional are interrelated by direct dependencies and correlate with one another This circumstance is similar to that encountered in the problem of constructing an approximant of a one-dimensional function from a discrete sampling involving data affected by considerable errors. A number of works of different authors have been recently published in which the minimization of the residual functional of the overdetermined systems of equations of the discrete problem is used in conventional versions of the FEM (see, e.g., [7]) and Galerkin techniques Their discussion goes beyond the scope of the given communication
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