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Application of Taylor-least squares finite element to three-dimensional advection-diffusion equation. (English) Zbl 0739.76033
Summary: The Taylor-least squares (TLS) scheme, developed to solve the unsteady advection-diffusion equation for advection-dominated cases in one and two dimensions, is extended to three dimensions and applied to some $$3D$$ examples to demonstrate its accuracy. The serendipity Hermite element is selected as an interpolation function on a linear hexagonal element. As a validation of the code and as a simple sensitivity analysis of the scheme on the different types of shape functions, the $$2D$$ example problem of the previous study is solved again. Four $$3D$$ problems, two with advection and two with advection-diffusion, are also solved. The first two examples are advection of a stepp $$3D$$ Gaussian hill in rotational flow fields. For the advection-diffusion problems with fairly general flow fields and diffusion tensors, analytical solutions are obtained using the ray method. Despite the steepness of the initial conditions, very good agreement is observed between the analytical and TLS solutions.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76R50 Diffusion
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##### References:
 [1] Park, Int. j. numer. methods fluids 11 pp 21– (1990) [2] Donea, J. Comput. Phys. 70 pp 463– (1987) [3] Carey, Int. j. numer. methods eng. 26 pp 81– (1988) [4] and , Numerical Solutions of Partial Differential Equations in Science and Engineering, Wiley-Interscience, New York, 1982. [5] ’Reference problems for the convection-diffusion forum’, Proc. VIth Int. Conf. on Finite Elements in Water Resources, Lisbon, Portugal, 1986. [6] Cohen, J. Inst. Math. Appl. 3 pp 266– (1967) [7] Smith, J. Fluid Mech. 111 pp 107– (1981) [8] ’Solutions of diffusion-advection problems by ray methods’, unpublished (see Reference 7). [9] , and , ’Triangular elements in plate bending–conforming and nonconforming solutions’, Proc. (First) Conf. on Matrix Methods in Structural Mechanics, AFFDL TR 66-80, 1965, pp. 547-576.
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