TY - JOUR

T1 - A combined FIC-TDG finite element approach for the numerical solution of coupled advection-diffusion-reaction equations with application to a bioregulatory model for bone fracture healing

AU - Sapotnick, A.

AU - Nackenhorst, U.

PY - 2012/10/19

Y1 - 2012/10/19

N2 - Numerical schemes for the approximative solution of advection-diffusion-reaction equations are often flawed because of spurious oscillations, caused by steep gradients or dominant advection or reaction. In addition, for strong coupled nonlinear processes, which may be described by a set of hyperbolic PDEs, established time stepping schemes lack either accuracy or stability to provide a reliable solution. In this contribution, an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques, namely the finite calculus (FIC-FEM) scheme introduced by Oñate etal. with time-discontinuous Galerkin (TDG) methods. Whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection-dominated problems, the latter ensures reliable solutions with regard to the temporal evolution. A brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests. The performance of the suggested FIC-TDG finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by Geris et al., which consists of at least 12 coupled hyperbolic evolution equations.

AB - Numerical schemes for the approximative solution of advection-diffusion-reaction equations are often flawed because of spurious oscillations, caused by steep gradients or dominant advection or reaction. In addition, for strong coupled nonlinear processes, which may be described by a set of hyperbolic PDEs, established time stepping schemes lack either accuracy or stability to provide a reliable solution. In this contribution, an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques, namely the finite calculus (FIC-FEM) scheme introduced by Oñate etal. with time-discontinuous Galerkin (TDG) methods. Whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection-dominated problems, the latter ensures reliable solutions with regard to the temporal evolution. A brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests. The performance of the suggested FIC-TDG finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by Geris et al., which consists of at least 12 coupled hyperbolic evolution equations.

KW - Advection

KW - Bone fracture healing

KW - Diffusion

KW - Finite calculus

KW - Finite element

KW - Hyperbolic PDE

KW - Reaction

KW - Time-discontinuous Galerkin

UR - http://www.scopus.com/inward/record.url?scp=84866849546&partnerID=8YFLogxK

U2 - 10.1002/nme.4338

DO - 10.1002/nme.4338

M3 - Article

AN - SCOPUS:84866849546

VL - 92

SP - 301

EP - 317

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 3

ER -