Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems

Robust Variational Physics-Informed Neural Networks

“…Lemma 8. Let u θ * be a quasi-minimizer and δ * > 0 be the smallest constant satisfying (38). It holds…”

“…Proposition 9 (Local a priori error estimate II). Let u θ * be a quasi-minimizer and δ * > 0 be the smallest constant satisfying (38). If Assumption 2 is satisfied, it holds:…”

“…The spirit of MinRes methods is to minimize the dual norm of the residual of the PDE given in variational form. Many strategies have been developed based on this idea over the last five decades, including the families of Galerkin Least-Squares methods [7,30,29], First Order Least-squares [12,13], residual minimization methods on dual discontinuous Galerkin norms [14,18,17,50,37], isogeometric residual minimization methods [15,39,38,40], and Discontinuous Petrov-Galerkin (DPG) methods [20,21,22,43,49]. As the residual operator lives in the dual of the test space and the dual norm is difficult to compute, the use of the Riesz representation theorem [45] is natural in this context.…”

Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

“…Mixing the advantages of the residual minimization with the benefits resulting from isogeometric analysis, implicit dynamics, and an alternating direction solver, a novel computational implicit method called Isogeometric Residual Minimization (iGRM) with direction splitting was recently introduced. Following this framework, an application of variational residual minimization to the stationary and timedependent Stokes equations is presented in [14]. B-splines are used to discretize both trial and test spaces, and the scheme is refactored in such a way as to expose Kronecker products.…”