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Marcus Sarkis

Publications and source records attributed to Marcus Sarkis.

13 recordsLinked to original sources
arXiv2026

Two-scale neural networks for optimal control of linear convection-dominated equations

We propose a two-scale neural network method for optimal control problems governed by convection-dominated convection-diffusion-reaction equations. Building on two-scale architectures developed for singularly perturbed forward problems, we augment the spatial input with suitably rescaled features that become increasingly important as the diffusion coefficient becomes small. The approach employs separate neural networks for the state and adjoint state variables of the optimality system, reflecting the fact that these quantities develop sharp layers in different parts of the domain due to opposite convection fields. By choosing different center points for the two networks, the architecture naturally aligns with the layer location of each variable. We present two formulations of the method, one based on the first-order optimality conditions and another using penalization of the PDE constraint, and combine them with a successive training strategy that gradually decreases the diffusion coefficient toward its target value. Numerical experiments on benchmark problems illustrate the effectiveness and behavior of the proposed approach.

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arXiv2025

Adaptive Nonoverlapping Preconditioners for the Helmholtz Equation

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel substructuring approach to mitigate the potential ill-posedness of local Dirichlet problems for the Helmholtz equation. We propose two types of preconditioners within the framework of nonoverlapping spectral additive Schwarz (NOSAS) methods. The first type of preconditioner focuses on the real part of the Helmholtz problem, while the second type addresses both the real and imaginary components, providing a comprehensive strategy to enhance scalability and reduce computational cost. Our approach is purely algebraic, which allows for adaptability to various discretizations and heterogeneous Helmholtz coefficients while maintaining theoretical convergence for thresholds close to zero. Numerical experiments confirm the effectiveness of the proposed preconditioners, demonstrating robust convergence rates and scalability, even for large wavenumbers.

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arXiv2024

Hybrid Localized Spectral Decomposition for multiscale problems

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.

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arXiv2021

From Additive Average Schwarz Methods to Non-overlapping Spectral Additive Schwarz Methods

In this paper, we design and analyze two new methods based on additive average Schwarz -- AAS method introduced in \cite{MR1943457}. The new methods design for elliptic problems with highly heterogeneous coefficients. The methods are of the non-overlapping type, and the subdomain interactions obtain via the coarse space. The first method is the minimum energy Schwarz -- MES method. MES has the minimum energy for the coarse space with constant extension inside each subdomain. The condition number of the MES method is always smaller than in the AAS method. The second class of methods is the non-overlapping spectral additive Schwarz -- NOSAS methods based on low-rank discrete energy harmonic extension in each subdomain. To achieve the low-rank, we solve a generalized eigenvalue problem in each subdomain. NOSAS have the minimum energy for a given rank of the coarse space. The condition number of the NOSAS methods does not depend on the coefficients. Additionally, the NOSAS methods have good parallelization properties. The size of the global problem is equal to the total number of eigenvalues chosen in each subdomain. It is only related to the number of high-permeable islands that touch the subdomains' interface.

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arXiv2021

Natural Factor Based Solvers

We consider parametric families of partial differential equations--PDEs where the parameter $κ$ modifies only the (1,1) block of a saddle point matrix product of a discretization below. The main goal is to develop an algorithm that removes, as much as possible, the dependence of iterative solvers on the parameter $κ$. The algorithm we propose requires only one matrix factorization which does not depend on $κ$, therefore, allows to reuse it for solving very fast a large number of discrete PDEs for different $κ$ and forcing terms. The design of the proposed algorithm is motivated by previous works on natural factor of formulation of the stiffness matrices and their stable numerical solvers. As an application, in two dimensions, we consider an iterative preconditioned solver based on the null space of Crouzeix-Raviart discrete gradient represented as the discrete curl of $P_1$ conforming finite element functions. For the numerical examples, we consider the case of random coefficient pressure equation where the permeability is modeled by an stochastic process. We note that contrarily from recycling Krylov subspace techniques, the proposed algorithm does not require fixed forcing terms.

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arXiv2018

An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid

In this paper we introduce a finite element method for the Stokes equations with a massless immersed membrane. This membrane applies normal and tangential forces affecting the velocity and pressure of the fluid. Additionally, the points representing this membrane move with the local fluid velocity. We design and implement a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane and then we formulate a time discretization that yields an unconditionally energy stable scheme. We prove that the stability is not restricted by the parameter choices that constrained previous finite element immersed boundary methods and illustrate the theoretical results with numerical simulations.

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arXiv2017

Spectral ACMS: A robust localized Approximated Component Mode Synthesis Method

We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into \emph{multiscale} and \emph{fine} subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$.

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arXiv2017

Analysis of the finite element method for the Laplace--Beltrami equation on surfaces with regions of high curvature using graded meshes

We derive error estimates for the piecewise linear finite element approximation of the Laplace--Beltrami operator on a bounded, orientable, $C^3$, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.

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arXiv2017

On high-order conservative finite element methods

A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred.

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arXiv2015

Higher-order finite element methods for elliptic problems with interfaces

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.

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arXiv2014

The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems

In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $\Omega$ which is a union of $N$ disjoint polygonal subdomains $\Omega_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces $\partial \Omega_i$. In each $\Omega_i$ a conforming quasiuniform triangulation with parameters $h_i$ is constructed. We assume that the resulting triangulation in $\Omega$ is also conforming, i.e., the meshes are assumed to match across the subdomain interfaces. On the fine triangulation the problem is discretized by a DG method. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by $C(1 + \max_i \log H_i/h_i)^2$ with a constant $C$ independent of $h_i$, $H_i$ and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. This result is an extension, to the case of full fine-grid DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013), pp.~400--422] where it was considered a conforming finite element method inside the subdomains and a discontinuous Galerkin method only across the subdomain interfaces. Numerical results are presented to validate the theory.

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