An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms

Abstract

We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.

Cited by in corpus (10)

• An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems
• A Deep Double Ritz Method (D${}^2$RM) for solving Partial Differential Equations using Neural Networks
• Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm
• A spatio-temporal adaptive phase-field fracture method
• Isogeometric Residual Minimization Method (iGRM) with Direction Splitting Preconditoner for Stationary Advection-Diffusion Problems
• A nonlinear weak constraint enforcement method for advection-dominated diffusion problems
• A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation
• Adaptive stabilized finite elements: Continuation analysis of compaction banding in geomaterials
• Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems
• An adaptive superconvergent finite element method based on local residual minimization