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The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of…
In this paper we develop two goal-oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework. In this methodology, one seeks to determine the number of multiscale basis…
The numerical approximation of convection-dominated problems continues to remain subject of strong interest. Families of stabilization techniques for finite element methods were developed in the past. Adaptive techniques based on a…
The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques…
This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element…
In this work, a space-time scheme for goal-oriented a posteriori error estimation is proposed. The error estimator is evaluated using a partition-of-unity dual-weighted residual method. As application, a low mach number combustion equation…
Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work,…
In this work, we consider multigoal-oriented error estimation for stationary fluid-structure interaction. The problem is formulated within a variational-monolithic setting using arbitrary Lagrangian-Eulerian coordinates. Employing the…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the…
In this work, we develop adaptive schemes using goal-oriented error control for a highly nonlinear flow temperature model with temperature dependent density. The dual-weighted residual method for computing error indicators to steer mesh…
In a dual weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have…
We consider the vertex-centered finite volume method with first-order conforming ansatz functions. The adaptive mesh-refinement is driven by the local contributions of the weighted-residual error estimator. We prove that the adaptive…
In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint…
This chapter describes how a posteriori error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique, can be easily applied for biomechanical simulations in current engineering practice.…
In this work, we implement goal-oriented error control and spatial mesh adaptivity for stationary fluid-structure interaction. The a posteriori error estimator is realized using the dual-weighted residual method in which the adjoint…
We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the…
We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable…
This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations,…
This article presents a general and novel approach to the automation of goal-oriented error control in the solution of nonlinear stationary finite element variational problems. The approach is based on automated linearization to obtain the…