Automatic Variationally Stable Analysis for FE Computations: An Introduction

Abstract

We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients. In the spirit of least squares FE methods (Bochev and Gunzburger, Least-Squares Finite Element Methods, vol 166, Springer Science & Business Media, Berlin, 2009), the AVS-FE method recasts the governing second order partial differential equation (PDE) into a system of first-order PDEs. However, in the subsequent derivation of the equivalent weak formulation, a Petrov-Galerkin technique is applied by using different regularities for the trial and test function spaces. We use standard FE approximation spaces for the trial spaces, which are C0, and broken Hilbert spaces for the test functions. Thus, we seek to compute pointwise continuous solutions for both the primal variable and its flux (as in least squares FE methods), while the test functions are piecewise discontinuous. To ensure the numerical stability of the subsequent FE discretizations, we apply the philosophy of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Comput Methods Appl Mech Eng 199(23):1558–1572, 2010; Discontinuous Petrov-Galerkin (DPG) method, Tech. rep., The Institute for Computational Engineering and Sciences, The University of Texas at Austin, 2015; SIAM J Numer Anal 49(5):1788–1809, 2011; Numer Methods Partial Differ Equ 27(1):70–105, 2011; Appl Numer Math 62(4):396–427,2012; Carstensen et al., SIAM J Numer Anal 52(3):1335–1353, 2014), by invoking test functions that lead to unconditionally stable numerical systems (if the kernel of the underlying differential operator is trivial). In the AVS-FE method, the discontinuous test functions are ascertained per the DPG approach from local, decoupled, and well-posed variational problems, which lead to best approximation properties in terms of the energy norm. We present various 2D numerical verifications, including convection-diffusion problems with highly oscillatory coefficients and extremely high Peclet numbers, up to O(109). These show the unconditional stability without the need for any upwind schemes nor any other artificial numerical stabilization. The results are not highly diffused for convection-dominated problems nor show any strong oscillations, but adequately capture and indicate the presence of boundary layers, even for very coarse meshes and low polynomial degrees of approximation, p. Remarkably, we can compute the test functions by using the same p level as the trial functions without significantly impacting the numerical accuracy or asymptotic convergence of the numerical results. In addition, the AVS method delivers high numerical accuracy for the computed flux. Importantly, the AVS methodology delivers optimal asymptotic error convergence rates of order p + 1 and p are obtained in the L2 and H1 norms for the primal variable. Our experience indicates that for convection-dominated problems we often observe a convergence rate of p + 1 for the L2 norm of the flux variable.