A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

by   Nuria Pares, et al.

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;Ω) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.



There are no comments yet.


page 16

page 18

page 19

page 20

page 22


A subcell-enriched Galerkin method for advection problems

In this work, we introduce a generalization of the enriched Galerkin (EG...

Goal-oriented adaptive mesh refinement for non-symmetric functional settings

In this article, a Petrov-Galerkin duality theory is developed. This the...

Agglomeration-Based Geometric Multigrid Solvers for Compact Discontinuous Galerkin Discretizations on Unstructured Meshes

We present a geometric multigrid solver for the Compact Discontinuous Ga...

Goal-oriented anisotropic hp-adaptive discontinuous Galerkin method for the Euler equations

We deal with the numerical solution of the compressible Euler equations ...

A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

In this paper, we present a hybridized discontinuous Galerkin (HDG) meth...

A Direct Sampling Method for the Inversion of the Radon Transform

We propose a novel direct sampling method (DSM) for the effective and st...

On the best constants in L^2 approximation

In this paper we provide explicit upper and lower bounds on the L^2n-wid...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.