Petia T. Boyanova

Efficient preconditioners for large scale binary Cahn–Hilliard models

Abstract In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the Cahn-Hilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the θ-method in time. The preconditioner utilises to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Various numerical experiments, including large scale examples, are presented as well as performance comparisons with other solution methods.

Numerical and computational efficiency of solvers for two-phase problems

We consider two-phase flow problems, modelled by the Cahn-Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential. We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn-Hilliard system to a problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation. We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.

Efficient numerical solution of discrete multi-component Cahn-Hilliard systems

In this work we develop preconditioners for the iterative solution of the large scale algebraic systems, arising in finite element discretizations of microstructures with an arbitrary number of components, described by the diffusive interface model. The suggested numerical techniques are applied to the study of ternary fluid flow processes.

Block-Preconditioners for conforming and non-conforming FEM discretizations of the cahn-hilliard equation

We consider preconditioned iterative solution methods to solve the algebraic systems of equations arising from finite element discretizations of multiphase flow problems, based on the phase-field model. The aim is to solve coupled physics problems, where both diffusive and convective processes take place simultaneously in time and space. To model the above, a coupled system of partial differential equations has to be solved, consisting of the Cahn-Hilliard equation to describe the diffusive interface and the time-dependent Navier-Stokes equation, to follow the evolution of the convection field in time. We focus on the construction and efficiency of preconditioned iterative solution methods for the linear systems, arising after conforming and non-conforming finite element discretizations of the Cahn-Hilliard equation in space and implicit discretization schemes in time. The non-linearity of the phase-separation process is treated by Newtons method. The resulting matrices admit a two-by-two block structure, utilized by the preconditioning techniques, proposed in the current work. We discuss approximation estimates of the preconditioners and include numerical experiments to illustrate their behaviour.

Robust AMLI methods for parabolic Crouzeix-Raviart FEM systems

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques. In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy-Bunyakowski-Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix. The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix-Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method.

Multilevel Splitting of Weighted Graph-Laplacian Arising in Non-conforming Mixed FEM Elliptic Problems

We consider a second-order elliptic problem in mixed form that has to be solved as a part of a projection algorithm for unsteady Navier-Stokes equations. The use of Crouzeix-Raviart non-conforming elements for the velocities and piece-wise constants for the pressure provides a locally mass-conservative algorithm. Then, the Crouzeix-Raviart mass matrix is diagonal, and the velocity unknowns can be eliminated exactly. The reduced matrix for the pressure is referred to as weighted graph-Laplacian. In this paper we study the construction of optimal order preconditioners based on algebraic multilevel iterations (AMLI). The weighted graph-Laplacian for the model 2-D problem is considered. We assume that the finest triangulation is obtained after recursive uniform refinement of a given coarse mesh. The introduced hierarchical splitting is the first important contribution of this article. The proposed construction allows for a local analysis of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. This is an important characteristic of the splitting and is associated with the angle between the two hierarchical FEM subspaces. The estimates of the convergence rate and the computational cost at each iteration show that the related AMLI algorithm with acceleration polynomial of degree two or three is of optimal complexity.

Multilevel preconditioning of graph-Laplacians

We consider the discrete system resulting from mixed finite element approximation of a second-order elliptic boundary value problem with Crouzeix-Raviart non-conforming elements for the vector valued unknown function and piece-wise constants for the scalar valued unknown function. Since the mass matrix corresponding to the vector valued variables is diagonal, these unknowns can be eliminated exactly. Thus, the problem of designing an efficient algorithm for the solution of the resulting algebraic system is reduced to one of constructing an efficient algorithm for a system whose matrix is a graph-Laplacian (or weighted graph-Laplacian).We propose a preconditioner based on an algebraic multilevel iterations (AMLI) algorithm. The hierarchical two-level transformations and the corresponding 2i?2 block splittings of the graph-Laplacian needed in an AMLI algorithm are introduced locally on macroelements. Each macroelement is associated with an edge of a coarser triangulation. To define the action of the preconditioner we employ polynomial approximations of the inverses of the pivot blocks in the 2i?2 splittings. Such approximations are obtained via the best polynomial approximation of x-1 in L∞ norm on a finite interval. Our construction provides sufficient accuracy and moreover, guarantees that each pivot block is approximated by a positive definite matrix polynomial.One possible application of the constructed efficient preconditioner is in the numerical solution of unsteady Navier-Stokes equations by a projection method. It can also be used to design efficient solvers for problems corresponding to other mixed finite element discretizations.

On Optimal AMLI Solvers for Incompressible Navier‐Stokes Problems

We consider the incompressible Navier‐Stokes problem and a projection scheme based on Crouzeix‐Raviart finite element approximation of the velocities and piece‐wise constant approximation of the pressure. These non‐conforming finite elements guarantee that the divergence of the velocity field is zero inside each element, i.e., the approximation is locally conservative.We propose optimal order Algebraic MultiLevel Iteration (AMLI) preconditioners for both, the decoupled scalar parabolic problems at the prediction step as well as to the mixed finite element method (FEM) problem at the projection step. The main contribution of the current paper is the obtained scalability of the AMLI methods for the related composite time‐stepping solution method. The algorithm for the Navier‐Stokes problem has a total computational complexity of optimal order. We present numerical tests for the efficiency of the AMLI solvers for the case of lid‐driven cavity flow for different Reynolds numbers.

Numerical study of AMLI methods for weighted graph-laplacians

We consider a second-order elliptic problem in mixed form that has to be solved as a part of a projection algorithm for unsteady Navier-Stokes equations The use of Crouzeix-Raviart non-conforming elements for the velocities and piece-wise constants for the pressure provides a locally mass-conservative approximation Since the mass matrix corresponding to the velocities is diagonal, these unknowns can be eliminated exactly We address the design of efficient solution methods for the reduced weighted graph-Laplacian system. Construction of preconditioners based on algebraic multilevel iterations (AMLI) is considered AMLI is a stabilized recursive generalization of two-level methods We define hierarchical two-level transformations and corresponding block 2x2 splittings locally for macroelements associated with the edges of the coarse triangulation Numerical results for two sets of hierarchical partitioning parameters are presented for the cases of two-level and AMLI methods The observed behavior complies with the theoretical expectations and the constructed AMLI preconditioners are optimal.