Giorgio Bornia

On the properties and limitations of the height function method in two-dimensional Cartesian geometry

In this study we define the continuous height function to investigate the approximation of an interface line and its geometrical properties with the height function method. We show that in each mixed cell the piecewise linear interface reconstruction and the approximation of the derivatives and curvature based on three consecutive height function values are second-order accurate. We also discuss the quadratic reconstruction and fourth-order accurate expressions of the normal and curvature. We present a hierarchical algorithm to compute the normal vector and curvature of an interface line with the height function method that switches automatically between second- and fourth-order approximations and that can be applied also when the local radius of curvature is of the order of the grid spacing.

FLUID-STRUCTURE SIMULATIONS AND BENCHMARKING OF ARTERY ANEURYSMS UNDER PULSATILE BLOOD FLOW

FLUID-STRUCTURE SIMULATIONS AND BENCHMARKING OF ARTERY ANEURYSMS UNDER PULSATILE BLOOD FLOW Eugenio Aulisa, Giorgio Bornia and Sara Calandrini 1 Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, USA e-mail: eugenio.aulisa@ttu.edu 2 Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, USA e-mail: giorgio.bornia@ttu.edu, sara.calandrini@ttu.edu

Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition

Abstract In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs under no regularity assumptions on the solution of the problem. The proposed analysis provides three main contributions to the existing theory. The first novel contribution of this study is a convergence bound that depends on the number of multigrid smoothing iterations. This result is obtained under no regularity assumptions on the solution of the problem. A similar result has been shown in the literature for the cases of full regularity and partial regularity assumptions. Second, our theory applies to local refinement applications with arbitrary level hanging nodes. More specifically, for the smoothing algorithm we provide subspace decompositions that are suitable for applications where the multigrid spaces are defined on finite element grids with arbitrary level hanging nodes. Third, global smoothing is employed on the entire multigrid space with hanging nodes. When hanging nodes are present, existing multigrid strategies advise to carry out the smoothing procedure only on a subspace of the multigrid space that does not contain hanging nodes. However, with such an approach, if the number of smoothing iterations is increased, convergence can improve only up to a saturation value. Global smoothing guarantees an arbitrary improvement in the convergence when the number of smoothing iterations is increased. Numerical results are also included to support our theoretical findings.

MULTIGRID SOLVER WITH DOMAIN DECOMPOSITION SMOOTHING FOR STEADY-STATE INCOMPRESSIBLE FSI PROBLEMS

In this paper we investigate the numerical performance of a monolithic Newtonmultigrid solver with domain decomposition smoothers for the solution of a class of stationary incompressible FSI problems. The physics of the problem is described using a monolithic approach, where mass continuity and stress balance are automatically satisfied across the fluidsolid interface. The deformation of the fluid domain is taken into account within the nonlinear Newton iterations according to an Arbitrary Lagrangian Eulerian (ALE) scheme. Due to the complexity and variety of the operators, the implementation of the Jacobian matrix in the nonlinear iterations is not a trivial task. To this purpose, we make use of automatic differentiation tools for an exact computation of the Jacobian matrix. The numerical solution of steady-state problems is particularly challenging, due to the ill-conditioning of the induced stiffness matrix. Moreover, the enforcement of the incompressibility condition calls for the use of incompressible solvers either of mixed or segregated type. At each nonlinear outer iteration the resulting linearized system is solved with a geometric multigrid solver. We consider a GMRES smoother preconditioned by an Additive Schwarz Method (ASM). The domain decomposition of the preconditioner is driven by the natural splitting between fluid and solid domain. The numerical results of some benchmark tests for steady-state cases show agreement with the literature and an increased robustness with our choice of smoothers with respect to standard ones.

Preface of the "Symposium on geometric methods for integrable systems and PDE with applications to engineering, biology and medicine"

The symposiums goal is to present recent advances in mathematical research of non-linear PDE and their immediate or long-term applications to the diverse engineering fields, as well as biology and medicine.