A Minimum-Residual Finite Element Method for the Convection-Diffusion Equation

Abstract : We present a minimum-residual finite element method for convection-diffusion problems in a higher order, adaptive, continuous Galerkin setting. The method borrows concepts from both the Discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [1] and the method of variational stabilization by Dahmen et al. [2], and it can also be interpreted as a variational multiscale method in which the fine-scales are defined through a dual-orthogonality condition. A key ingredient in the method is the proper choice of norm used to measure the residual, and we present two alternatives which are observed to be robust in both convection and diffusion-dominated regimes. Numerically obtained convergence rates are given in 2D, and benchmark numerical examples in all space dimensions are shown to illustrate the behavior of the method.

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