Related papers: Goal-oriented adaptivity for GMsFEM
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…
The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with…
This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators, that are…
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…
The cost- and memory-efficient numerical simulation of coupled volume-based multi-physics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goal-oriented space…
We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [SIAM J. Sci. Comput.,…
We propose a new heuristic goal-oriented a posteriori error estimator that connects the dual weighted residual method with equilibrated a posteriori error estimation. Our numerical experiments demonstrate the practical reliability of the…
Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the…
In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency…
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 introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for…
Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators,…
In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which…
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…
Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the…
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…
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…
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by…
We propose and analyze a general goal-oriented adaptive strategy for approximating quantities of interest (QoIs) associated with solutions to linear elliptic partial differential equations with random inputs. The QoIs are represented by…