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A discontinuous \(hp\) finite element method for diffusion problems: 1-D analysis. (English) Zbl 0940.65076
The mathematical background for a new type of discontinuous Galerkin method is reported, including stability analysis and error estimates, and is illustrated by the numerical analysis of a steady state 1-D diffusion problem. The shape functions are discontinuous at the element boundaries. The main features of the method are: auxiliary variables are not needed; robustness; the method yields elementwise conservative approximations; resulting matrices are positive-definite, well conditioned and convenient for the treatment of transient problems; adaptive error control; acceptable costs.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65L20 Stability and convergence of numerical methods 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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