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On the use of 'arc length' and 'defect' for mesh selection for differential equations

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In this note we will consider mesh selection techniques for ordinary differential equations (ODEs) and for partial differential equations (PDEs) based on arc length and defect. We assume we are trying to approximate y(x) for x ∈ [0, T] in the ODE case and u(x, y) for (x, y) ∈ Ω in the PDE case. The two specific areas of application that we have in mind are the numerical solution of boundary value problems (BVPs) in ODEs and the numerical solution of PDEs when a method of lines (MOL) approach is employed. The approach we develop is applicable to a wider class of PDEs including mixed-order problems and 3D problems.


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