Title: High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Abstract

Here, we propose an extension of the embedded boundary method known as “shifted boundary method” to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, thesemore » two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.« less

Authors:

Nouveau, L. ^[1]; Ricchiuto, M. ^[2]; Scovazzi, G.  ^[1]

• Search DOE PAGES for author "Scovazzi, G."
• Search DOE PAGES for ORCID "0000-0003-1043-6281"

---

+ Show Author Affiliations

1. Duke Univ., Durham, NC (United States)
2. INRIA, Bordeaux (France)

Publication Date:

Thu Aug 22 00:00:00 EDT 2019

Research Org.:

Duke Univ., Durham, NC (United States)

Sponsoring Org.:

USDOE Office of Science (SC); US Army Research Office (ARO)

OSTI Identifier:

1802315

Alternate Identifier(s):

OSTI ID: 1561375

Grant/Contract Number:  

SC0012169; W911NF1810308

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 398; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; computer science; physics; darcy flow; embedded boundary; finite element method; high-order approximation; stabilized methods; computational fluid dynamics