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Papers by Maria Dimitrova Lymbery

Mathematics of Computation

In this paper we prove a new abstract stability result for perturbed saddle-point problems based ... more In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuška’s theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, and the other standard assumptions in Brezzi’s theory, in a combined abstract norm. The construction suggests to form the latter from individual fitted norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot’s equations.

Numerical Linear Algebra With Applications, Oct 14, 2014

In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The constructio... more In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The construction is based on a two-level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non-overlapping subdomains. The two-level method utilizes a coarse-grid operator obtained from additive Schur complement approximation (ASCA). Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both, the two-level preconditioner, as well as for the ASCA are derived. The two-level method is recursively extended to define the ASMG algorithm. In particular, so-called Krylov-cycles are considered. The theoretical results are supported by a representative collection of numerical tests which further demonstrate the efficiency of the new algorithm for multiscale problems.

arXiv (Cornell University), May 13, 2022

Numerical Linear Algebra With Applications, Apr 3, 2019

The parameters in the governing system of partial differential equations of multicompartmental po... more The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ, but also with respect to all the other model parameters such as permeability coefficients Ki, storage coefficients cp i , network transfer coefficients βij , i, j = 1, • • • , n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameterrobust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.

arXiv (Cornell University), Oct 13, 2019

Computer Methods in Applied Mechanics and Engineering, Oct 1, 2021

arXiv (Cornell University), Mar 16, 2021

arXiv (Cornell University), May 30, 2018

arXiv (Cornell University), Jul 8, 2021

Journal of Scientific Computing

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an ela... more The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

Computers & Mathematics with Applications

arXiv (Cornell University), Mar 16, 2021

Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, 2013

We present an overview on the state of the art of robust AMLI preconditioners for anisotropic ell... more We present an overview on the state of the art of robust AMLI preconditioners for anisotropic elliptic problems. The included theoretical results summarize the convergence analysis of both linear and nonlinear AMLI methods for finite element discretizations by conforming and nonconforming linear elements and by conforming quadratic elements. The initially proposed hierarchical basis approach leads to robust multilevel algorithms for linear but not for quadratic elements for which an alternative AMLI method based on additive Schur complement approximation (ASCA) has been developed by the authors just recently. The presented new numerical results are focused on cases beyond the limitations of the rigorous AMLI theory. They reveal the potential and prospects of the ASCA approach to enhance the robustness of the resulting AMLI methods especially in situations when the matrix-valued coefficient function is not resolved on the coarsest mesh in the multilevel hierarchy.

Large-Scale Scientific Computing, 2012

Computer Methods in Applied Mechanics and Engineering, 2021

Multiscale Modeling & Simulation, 2020

SIAM Journal on Scientific Computing, 2016

In this paper we propose and analyze a preconditioner for a system arising from a mixed finite el... more In this paper we propose and analyze a preconditioner for a system arising from a mixed finite element approximation of second-order elliptic problems describing processes in highly heterogeneous media. Our approach uses the technique of multilevel methods (see, e.g., [P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations, Springer, New York, 2008]) and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus [SIAM J. Sci. Comput., 34 (2012), pp. A2872--A2895]. The main results are the design, study, and numerical justification of iterative algorithms for these problems that are robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient of the problem. Numerical tests provide experimental evidence for the high quality of the preconditioner and its desired robustness with respect to the material contrast. Such resu...

Numerical Linear Algebra with Applications, 2014

SummaryIn this paper, the idea of auxiliary space multigrid methods is introduced. The constructi... more SummaryIn this paper, the idea of auxiliary space multigrid methods is introduced. The construction is based on a two‐level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non‐overlapping subdomains. The two‐level method utilizes a coarse‐grid operator obtained from additive Schur complement approximation. Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both the two‐level preconditioner and the additive Schur complement approximation are derived. The two‐level method is recursively extended to define the auxiliary space multigrid algorithm. In particular, so‐called Krylov cycles are considered. The theoretical results are supported by a representative collection of numerical tests that further demonstrate the efficiency of the new algorithm for multiscale problems. Copyright © 2014 John Wiley & Sons, Ltd.

Mathematics of Computation

In this paper we prove a new abstract stability result for perturbed saddle-point problems based ... more In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuška’s theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, and the other standard assumptions in Brezzi’s theory, in a combined abstract norm. The construction suggests to form the latter from individual fitted norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot’s equations.

Numerical Linear Algebra With Applications, Oct 14, 2014

In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The constructio... more In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The construction is based on a two-level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non-overlapping subdomains. The two-level method utilizes a coarse-grid operator obtained from additive Schur complement approximation (ASCA). Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both, the two-level preconditioner, as well as for the ASCA are derived. The two-level method is recursively extended to define the ASMG algorithm. In particular, so-called Krylov-cycles are considered. The theoretical results are supported by a representative collection of numerical tests which further demonstrate the efficiency of the new algorithm for multiscale problems.

arXiv (Cornell University), May 13, 2022

Numerical Linear Algebra With Applications, Apr 3, 2019

The parameters in the governing system of partial differential equations of multicompartmental po... more The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ, but also with respect to all the other model parameters such as permeability coefficients Ki, storage coefficients cp i , network transfer coefficients βij , i, j = 1, • • • , n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameterrobust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.

arXiv (Cornell University), Oct 13, 2019

Computer Methods in Applied Mechanics and Engineering, Oct 1, 2021

arXiv (Cornell University), Mar 16, 2021

arXiv (Cornell University), May 30, 2018

arXiv (Cornell University), Jul 8, 2021

Journal of Scientific Computing

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an ela... more The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

Computers & Mathematics with Applications

arXiv (Cornell University), Mar 16, 2021

Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, 2013

We present an overview on the state of the art of robust AMLI preconditioners for anisotropic ell... more We present an overview on the state of the art of robust AMLI preconditioners for anisotropic elliptic problems. The included theoretical results summarize the convergence analysis of both linear and nonlinear AMLI methods for finite element discretizations by conforming and nonconforming linear elements and by conforming quadratic elements. The initially proposed hierarchical basis approach leads to robust multilevel algorithms for linear but not for quadratic elements for which an alternative AMLI method based on additive Schur complement approximation (ASCA) has been developed by the authors just recently. The presented new numerical results are focused on cases beyond the limitations of the rigorous AMLI theory. They reveal the potential and prospects of the ASCA approach to enhance the robustness of the resulting AMLI methods especially in situations when the matrix-valued coefficient function is not resolved on the coarsest mesh in the multilevel hierarchy.

Large-Scale Scientific Computing, 2012

Computer Methods in Applied Mechanics and Engineering, 2021

Multiscale Modeling & Simulation, 2020

SIAM Journal on Scientific Computing, 2016

In this paper we propose and analyze a preconditioner for a system arising from a mixed finite el... more In this paper we propose and analyze a preconditioner for a system arising from a mixed finite element approximation of second-order elliptic problems describing processes in highly heterogeneous media. Our approach uses the technique of multilevel methods (see, e.g., [P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations, Springer, New York, 2008]) and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus [SIAM J. Sci. Comput., 34 (2012), pp. A2872--A2895]. The main results are the design, study, and numerical justification of iterative algorithms for these problems that are robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient of the problem. Numerical tests provide experimental evidence for the high quality of the preconditioner and its desired robustness with respect to the material contrast. Such resu...

Numerical Linear Algebra with Applications, 2014

SummaryIn this paper, the idea of auxiliary space multigrid methods is introduced. The constructi... more SummaryIn this paper, the idea of auxiliary space multigrid methods is introduced. The construction is based on a two‐level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non‐overlapping subdomains. The two‐level method utilizes a coarse‐grid operator obtained from additive Schur complement approximation. Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both the two‐level preconditioner and the additive Schur complement approximation are derived. The two‐level method is recursively extended to define the auxiliary space multigrid algorithm. In particular, so‐called Krylov cycles are considered. The theoretical results are supported by a representative collection of numerical tests that further demonstrate the efficiency of the new algorithm for multiscale problems. Copyright © 2014 John Wiley & Sons, Ltd.