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A parameter-free stabilized finite element method for scalar advection-diffusion problems

100%
Bochev P., Peterson K.
Open Mathematics
|
2013
|
tom 11
|
nr 8
1458-1477
EN
We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.
2
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The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

76%
Lustyk M., Janus J., Pytel-Kudela M., Prykarpatsky A.
Open Mathematics
|
2009
|
tom 7
|
nr 4
775-786
EN
The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained.
3
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Anisotropic interpolation error estimates via orthogonal expansions

64%
Li M., Mao S.
Open Mathematics
|
2013
|
tom 11
|
nr 4
621-629
EN
We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.
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