Research

Research areas:

finite element method, parametric PDEs, subspace interpolation, iterative methods, mixed methods

Publications:

•  Results on a Mixed Finite Element Approach for a Model Convection-Diffusion Problem, with Daniel Hayes and Tyler O’Grady, to appear in Applicable Analysis. PDF
• Efficient discretization and preconditioning of the singularly perturbed Reaction-Diffusion problem, with Daniel Hayes and Jacob Jacavage, Computers and Mathematics with Applications, Volume 109, 270–279, 2022.
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• Connections Between Finite Difference and Finite Element Approximations, with Cristina Bacuta, Applicable Analysis, Vol. 102, no. 6, 2021, (arXiv:2107.06965, July 14, 2021).
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• Notes on a saddle point reformulation of mixed variational problems, with Daniel Hayes and Jacob Jacavage, Computers and Mathematics with Applications, Volume 95, 4–18, 2021.
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• Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms, with with L. Demkowicz, J. Mora Paz, and C. Xenophontos, Computers and Mathematics with Applications, 2020
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• Notes on a saddle point reformulation of mixed variational problems, with Daniel Hayes and Jacob Jacavage, Computers and Mathematics with Applications, 2020
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• Saddle Point Least Squares for the Reaction Diffusion Problem, with Jacob Jacavage, Results in Applied Mathematics, 2020
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• Least squares preconditioning for mixed methods with nonconforming trial spaces, with Jacob Jacavage, Applicable Analysis, 2019
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• A Non-conforming Saddle Point Least Squares Approach for Elliptic Interface Problems, with Jacob Jacavage, Computational Methods in Applied Mathematics, Volume 19: Issue 3, 399–414, 2019
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• Saddle Point Least Squares Preconditioning of Mixed Methods, with Jacob Jacavage, Computers and Mathematics with Applications, Volume 77, no. 5, 1396–1407, 2019.
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• Saddle point least squares iterative solvers for the time harmonic Maxwell equations, with Francisco-Javier Sayas, Jacob Jacavage, and Klajdi Qirko, Computers and Mathematics with Applications, 74, no. 11, 2915–2928, 2017.
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• Differential operators on domains with conical points: precise uniform regularity estimates, with Victor Nistor and Hengguang Li, Revue Roumanie de Mathematique Pures et Appliques, Vol. 62, no. 3, 383–411, 2017.
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• A saddle point least squares approach for primal mixed formulations of second order PDEs, with Klajdi Qirko, Computers and Mathematics with Applications, Volume 73, No. 2, 173–186, 2017.
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• Sharp Stability and Approximation Estimates for Symmetric Saddle Point Systems, Applicable Analysis, Volume 95, No. 1, 226–237, 2016.
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• A Saddle Point Least Squares approach to Mixed Methods, with Klajdi Qirko, Computers and Mathematics with Applications, Volume 70, Issue 12, pp 2920–2932, 2015.
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• Cascadic Multilevel Algorithms for Symmetric Saddle Point Systems, Computers and Mathematics with Applications, Volume 67, Issue 10, pp 1905–1913, 2014.
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• Equidistribution and Optimal Approximation Class, with Long Chen, and Jinchao Xu, book chapter, Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, Volume 91, pp 3-14, 2014.
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• Multilevel Gradient Uzawa Algorithms for Symmetric Saddle Point Problems, with Lu Shu, Jornal of Scientific Computing, 2013,Volume 57, Issue 1, pp 105-123, 2013
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• Anisotropic regularity and optimal rates of convergence for the Finite Element Method on three dimensional polyhedral domains, with A. Mazzucato, V. Nistor, and L. Zikatanov, book chapter in Advances in Mathematics, Bucharest Academy, pp 57-73, 2013.
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• Multilevel discretization of symmetric saddle point systems without the discrete LBB condition, with Peter Monk, Applied Numerical Mathematics, Volume 62, Issue 6, pp 667-681, 2012.
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• Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D, with Victor Nistor, and Hengguang Li, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS, 2012.
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• Residual Reduction Algorithms for Nonsymmetric Saddle Point Problems, with B. McCracken, and l. Shu, Journal of Computational and Applied Mathematics, Volume 235, Issue 6, pp 1614-1628, 2012.
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• Partition of Unity Refinement for local approximation, with Jiguang Sun and C. Zheng, Numerical Methods for Partial Differential Equations, Volume 27(4), pp 803-817, 2011.
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• A new approach for solving Stokes systems arising from a distributive relaxation method, with P. Vassilevski, and S. Zhang, Numerical Methods for Partial Differential Equations, Volume 27(4), pp 898-914, 2011
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• Interface and mixed boundary value problems on n-dimensional polyhedral domains, with A. Mazzucato, V. Nistor, and L. Zikatanov, Documenta Mathematica 15, 687–745, 2010.
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• Notes on the Schwartz Alternating Method for Partition of Unity FEM, with Jiguang Sun, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, Vol. 16(S1), pp 15-21, 2009.
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• Schur Complements on Hilbert Spaces and Saddle Point Systems, Journal of Computational and Applied Mathematics, Volume 225, Issue 2, pp 581-593, 2009.
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• Subspace interpolation with applications to elliptic regularity, Numerical Functional Analysis and Optimization, Volume 29, Issue 1&2, 2008, pp 88-114.
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• Improving the rate of convergence of `high order finite elements’ on polyhedra II: Mesh Refinements and Interpolation, with V.~Nistor and L.~Zikatanov, Numerical Functional Analysis and Optimization, Vol. 28, No 7-8, 2007, pp. 775-824.
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• A Unified Approach for Uzawa Algorithms, SIAM Journal of Numerical Analysis, Volume 44, Issue 6, 2006, pp. 2633-2649
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• Partition of Unity Finite Element Method Implementation for Poisson Equation, with J. Sun, Advances in Applied and Computational Mathematics, Nova Publishers, 2006, pp. 35-46
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• Improving the rate of convergence of `high order finite elements’ on polyhedra I: apriori estimates, with V.~Nistor and L.~Zikatanov, Numerical Functional Analysis and Optimization, Vol. 26, No 6, 2005, pp. 613-639.
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• Improving the rate of convergence of `high order finite elements’ on polygons and domains with cusps, with V. Nistor and L. Zikatanov, Numerische Mathematik, Vol. 100, No 2, 2005, pp. 165 -184
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• Partition of unity method on non-matching grids for the Stokes problem, with J.~Chen, Y.~Huang, J.~Xu and L.T.~Zikatanov, Journal of Numerical Mathematics, Vol.13, No.3, 2005, pp.157-169.
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• Regularity estimates for elliptic boundary value problems in Besov spaces, with J. H. Bramble and Jinchao Xu, Mathematics of Computation, 72 (2003), 1577-1595.
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• Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains, with J. H. Bramble and Jinchao Xu, Jornal of Numerical Mathematics, Vol.11, No.2, 2003, pp.75-177.
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• Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains with J. H. Bramble, Z. angew. Math. Phys.(ZAMP), 54(2003), 874-878.
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• Using finite element tools in proving shift theorems for elliptic boundary value problems, with J.H. Bramble, and J.E. Pasciak, Numerical Linear Algebra with Application, Volume 10, Issue no 1-2, 33-64, 2003.
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• New Interpolation Results and Applications to Finite Element Methods for Elliptic Boundary Value Problems, with J. H. Bramble and J. Pasciak. East-West J. Numer. Math. Vol.9, No.3, 2001, pp. 1-20.
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• Shift theorems for the biharmonic Dirichlet problem, with J. H. Bramble and J. Pasciak, Recent Progress in Computational and Applied PDEs by Kluwer Academic/Plenum Publishers, volume of proceedings for the CAPDE conference held in Zhangjiajie, China, 2001, Kluwer/Plenum, New York, pp. 1–26, 2002.
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• Boundary value problems and regularity on polyhedral domains, with V.~Nistor and L.~Zikatanov, IMA Preprint Series # 1984, August 20, 2004.
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• Interpolation between subspaces of Hilbert spaces and applications to shift theorems for elliptic boundary value problems and finite element methods. Thesis (Ph.D.) Texas A&M University, 2000.
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