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On an \(h\)-\(p\) version of the finite element method for a one-dimensional boundary value problem with singularity of a solution. (English) Zbl 0906.65084
Authors’ summary: The paper analyzes an \(h\)-\(p\) version of the finite element method for a one-dimensional test boundary value problem with coordinated degeneration of initial data and with strong singularity of a solution. The scheme of the finite element method is constructed on the basis of the definition of an \(R_{\nu }\)-generalized solution to the problem; the finite element space contains singular power functions. By using meshes with concentration at a singular point and by constructing the linear degree vector of approximating functions in a special way, a nearly optimal two-sided exponential estimate is obtained for the residual of the finite element method.
MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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