SUMMARYIn this paper we develop an adaptive finite element method for heat transfer in incompressible fluid flow. The adaptive method is based on an a posteriori error estimate for the coupled problem, which identifies how accurately the flow and heat transfer problems must be solved in order to achieve overall accuracy in a specified goal quantity. The a posteriori error estimate is derived using duality techniques and is of dual weighted residual type. We consider, in particular, an a posteriori error estimate for a variational approximation of the integrated heat flux through the boundary of a hot object immersed into a cooling fluid flow. We illustrate the method on some test cases involving three-dimensional time-dependent flow and transport.
In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.
In this paper we present the adaptive variational multiscale method for solving the Poisson equation in mixed form. We use the method introduced in [3], and further analyzed and applied to mixed problems in [4], which is a general tool for solving linear partial differential equations with multiscale features in the coefficients. We extend the numerics in [4] from rectangular meshes to triangular meshes which allow for computation on more complicated domains. A new a posteriori error estimate is also included, which is used in an adaptive algorithm. We present a numerical example that shows the efficiency of incorporating a posteriori based adaptivity into the method.
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