Numerische Mathematik, ISSN 0029-599X, 03/2016, Volume 132, Issue 3, pp. 519 - 539

This paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in H(div) and displacements in...

65N15 | 65N12 | 65N30 | ELASTICITY | EQUATIONS | MATHEMATICS, APPLIED | UNIFYING THEORY

65N15 | 65N12 | 65N30 | ELASTICITY | EQUATIONS | MATHEMATICS, APPLIED | UNIFYING THEORY

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 6/2019, Volume 79, Issue 3, pp. 1572 - 1607

For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important....

Discontinuous Galerkin | 65N15 | Computational Mathematics and Numerical Analysis | 65N12 | Error analysis | 65N35 | Theoretical, Mathematical and Computational Physics | Summation-by-parts | 65N06 | Error bound | Mathematics | Algorithms | Flux reconstruction | Mathematical and Computational Engineering | Error growth | 65N30 | BY-PARTS OPERATORS | MATHEMATICS, APPLIED | SPLIT-FORM | CONSERVATION | Analysis | Environmental law | Methods | Differential equations | Mathematics - Numerical Analysis

Discontinuous Galerkin | 65N15 | Computational Mathematics and Numerical Analysis | 65N12 | Error analysis | 65N35 | Theoretical, Mathematical and Computational Physics | Summation-by-parts | 65N06 | Error bound | Mathematics | Algorithms | Flux reconstruction | Mathematical and Computational Engineering | Error growth | 65N30 | BY-PARTS OPERATORS | MATHEMATICS, APPLIED | SPLIT-FORM | CONSERVATION | Analysis | Environmental law | Methods | Differential equations | Mathematics - Numerical Analysis

Journal Article

Numerische Mathematik, ISSN 0029-599X, 12/2017, Volume 137, Issue 4, pp. 857 - 893

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of...

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | 65G99 | Numerical and Computational Physics, Simulation | 65N30 | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | DISCONTINUOUS GALERKIN METHODS

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | 65G99 | Numerical and Computational Physics, Simulation | 65N30 | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | DISCONTINUOUS GALERKIN METHODS

Journal Article

Journal of computational methods in applied mathematics, ISSN 1609-9389, 2019, Volume 20, Issue 3, pp. 437 - 458

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear...

Linear Scalar Hyperbolic Equation | Gradient Discretisation Method | 65N12 | Numerical Tests | Convergence Analysis | 65N30

Linear Scalar Hyperbolic Equation | Gradient Discretisation Method | 65N12 | Numerical Tests | Convergence Analysis | 65N30

Journal Article

Numerische Mathematik, ISSN 0029-599X, 09/2016, Volume 134, Issue 1, pp. 197 - 222

A novel residual-type a posteriori error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or...

65N15 | 76S05 | 65N12 | 65N30 | 65N06 | MATHEMATICS, APPLIED | CONVERGENCE | APPROXIMATIONS | DISCRETIZATIONS | Finite element method | Analysis | Methods

65N15 | 76S05 | 65N12 | 65N30 | 65N06 | MATHEMATICS, APPLIED | CONVERGENCE | APPROXIMATIONS | DISCRETIZATIONS | Finite element method | Analysis | Methods

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 09/2016, Volume 309, pp. 579 - 609

The flexibility of the DPG methodology is exposed by solving the linear elasticity equations under different variational formulations, including some with...

Adaptivity | DPG | Linear elasticity | Variational formulations | Minimum residual method | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | WEAKLY IMPOSED SYMMETRY | FINITE-ELEMENT METHODS | EQUATIONS | Mathematics - Numerical Analysis

Adaptivity | DPG | Linear elasticity | Variational formulations | Minimum residual method | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | WEAKLY IMPOSED SYMMETRY | FINITE-ELEMENT METHODS | EQUATIONS | Mathematics - Numerical Analysis

Journal Article

Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, 07/2014, Volume 24, Issue 8, pp. 1575 - 1619

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the...

Elliptic equation | Discrete duality finite volume schemes | Coercivity | Minimum and maximum principles | Multi-point flux approximation | Review | Monotony | Finite volume schemes | Hybrid mimetic mixed methods | Convergence analysis | DISCRETE DUALITY | MATHEMATICS, APPLIED | coercivity | hybrid mimetic mixed methods | multi-point flux approximation | monotony | GENERAL 2D MESHES | convergence analysis | MULTIPOINT FLUX APPROXIMATION | QUADRILATERAL GRIDS | TENSOR PRESSURE EQUATION | DIFFERENCE METHOD | discrete duality finite volume schemes | UNSTRUCTURED POLYHEDRAL MESHES | NONLINEAR ELLIPTIC-EQUATIONS | POLYGONAL MESHES | finite volume schemes | minimum and maximum principles | CENTERED GALERKIN METHODS | elliptic equation | Mathematics - Numerical Analysis

Elliptic equation | Discrete duality finite volume schemes | Coercivity | Minimum and maximum principles | Multi-point flux approximation | Review | Monotony | Finite volume schemes | Hybrid mimetic mixed methods | Convergence analysis | DISCRETE DUALITY | MATHEMATICS, APPLIED | coercivity | hybrid mimetic mixed methods | multi-point flux approximation | monotony | GENERAL 2D MESHES | convergence analysis | MULTIPOINT FLUX APPROXIMATION | QUADRILATERAL GRIDS | TENSOR PRESSURE EQUATION | DIFFERENCE METHOD | discrete duality finite volume schemes | UNSTRUCTURED POLYHEDRAL MESHES | NONLINEAR ELLIPTIC-EQUATIONS | POLYGONAL MESHES | finite volume schemes | minimum and maximum principles | CENTERED GALERKIN METHODS | elliptic equation | Mathematics - Numerical Analysis

Journal Article

Numerische Mathematik, ISSN 0029-599X, 5/2019, Volume 142, Issue 1, pp. 1 - 32

We generalize the two dimensional mixed finite elements of Arbogast and Correa (SIAM J Numer Anal 54:3332–3356, 2016) defined on quadrilaterals to three...

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 41A10 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 41A10 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30

Journal Article

9.
Full Text
A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation

Calcolo, ISSN 1126-5434, 2018, Volume 55, Issue 3, pp. 1 - 39

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation problems in weak formulation. We...

65N15 | 65N12 | Aubin–Nitsche trick | Virtual element methods | 65N08 | Consistency | Mathematics | Theory of Computation | Strang lemma | Finite volume methods | Numerical Analysis | Error estimate | Oblique elliptic projector | 65N30 | MATHEMATICS, APPLIED | APPROXIMATIONS | Aubin-Nitsche trick | MATHEMATICS | DISCRETIZATION | ELEMENT METHODS | ANISOTROPIC DIFFUSION | Projectors | Anisotropy | Analysis | Error analysis | Approximation | Partial differential equations | Mathematical analysis | Dependence | Finite volume method | Fluxes | Estimates | Mathematics - Numerical Analysis

65N15 | 65N12 | Aubin–Nitsche trick | Virtual element methods | 65N08 | Consistency | Mathematics | Theory of Computation | Strang lemma | Finite volume methods | Numerical Analysis | Error estimate | Oblique elliptic projector | 65N30 | MATHEMATICS, APPLIED | APPROXIMATIONS | Aubin-Nitsche trick | MATHEMATICS | DISCRETIZATION | ELEMENT METHODS | ANISOTROPIC DIFFUSION | Projectors | Anisotropy | Analysis | Error analysis | Approximation | Partial differential equations | Mathematical analysis | Dependence | Finite volume method | Fluxes | Estimates | Mathematics - Numerical Analysis

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 8/2015, Volume 64, Issue 2, pp. 559 - 585

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is...

65N15 | Computational Mathematics and Numerical Analysis | 65N12 | Secondary: 35B45 | 35J50 | Theoretical, Mathematical and Computational Physics | Weak Laplacian | Mathematics | 35J35 | Weak Galerkin finite element methods | 74N20 | Algorithms | Primary: 65N30 | Appl.Mathematics/Computational Methods of Engineering | Polyhedral meshes | Biharmonic equation | MIXED METHOD | MATHEMATICS, APPLIED | 2ND-ORDER ELLIPTIC PROBLEMS | APPROXIMATIONS | FORMULATION | RECTANGLES | FLOW | NUMERICAL-SOLUTION | DIRICHLET PROBLEM | DOMAINS | COLLOCATION

65N15 | Computational Mathematics and Numerical Analysis | 65N12 | Secondary: 35B45 | 35J50 | Theoretical, Mathematical and Computational Physics | Weak Laplacian | Mathematics | 35J35 | Weak Galerkin finite element methods | 74N20 | Algorithms | Primary: 65N30 | Appl.Mathematics/Computational Methods of Engineering | Polyhedral meshes | Biharmonic equation | MIXED METHOD | MATHEMATICS, APPLIED | 2ND-ORDER ELLIPTIC PROBLEMS | APPROXIMATIONS | FORMULATION | RECTANGLES | FLOW | NUMERICAL-SOLUTION | DIRICHLET PROBLEM | DOMAINS | COLLOCATION

Journal Article

Numerische Mathematik, ISSN 0945-3245, 2019, Volume 143, Issue 1, pp. 139 - 175

An explicit and computable error estimator for the $$hp$$ hp version of the virtual element method (VEM), together with lower and upper bounds with respect to...

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | MATHEMATICS, APPLIED | MATHEMATICS AND COMPUTING

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | MATHEMATICS, APPLIED | MATHEMATICS AND COMPUTING

Journal Article

Comptes rendus - Mathématique, ISSN 1631-073X, 12/2016, Volume 354, Issue 12, pp. 1236 - 1240

In this article we introduce new possibilities of bounding the stability constants that play a vital role in the reduced basis method. By bounding stability...

MATHEMATICS

MATHEMATICS

Journal Article

Numerische Mathematik, ISSN 0029-599X, 6/2016, Volume 133, Issue 2, pp. 203 - 231

We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and $$L^2$$ L...

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N30 | Finite element method | Mechanical engineering | Analysis | Methods

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N30 | Finite element method | Mechanical engineering | Analysis | Methods

Journal Article

14.
Full Text
A fictitious domain approach with Lagrange multiplier for fluid-structure interactions

Numerische Mathematik, ISSN 0945-3245, 2016, Volume 135, Issue 3, pp. 711 - 732

We study a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the...

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | 74F10 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | MATHEMATICS, APPLIED | L-2 PROJECTION | FINITE-ELEMENT SPACES | IMMERSED BOUNDARY METHOD | STABILITY

Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | 74F10 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | MATHEMATICS, APPLIED | L-2 PROJECTION | FINITE-ELEMENT SPACES | IMMERSED BOUNDARY METHOD | STABILITY

Journal Article

Computer methods in applied mechanics and engineering, ISSN 0045-7825, 2016, Volume 309, pp. 497 - 531

We develop a model describing the behavior of two-phase ferrofluid flows using phase field-techniques and present an energy-stable numerical scheme for it. For...

Magnetization | Microstructure | Ferrofluids | Two-phase flow | Incompressible flows | FERROMAGNETIC FLUID | INCOMPRESSIBLE FLUIDS | MAGNETIC-FIELDS | CONVEX POLYHEDRA | NUMERICAL APPROXIMATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | NAVIER-STOKES EQUATIONS | ENGINEERING, MULTIDISCIPLINARY | ERROR ANALYSIS | FINITE-ELEMENT APPROXIMATIONS | CAHN-HILLIARD EQUATIONS | GLOBAL WEAK SOLUTIONS | Ferroalloys | Analysis | Mathematics - Numerical Analysis

Magnetization | Microstructure | Ferrofluids | Two-phase flow | Incompressible flows | FERROMAGNETIC FLUID | INCOMPRESSIBLE FLUIDS | MAGNETIC-FIELDS | CONVEX POLYHEDRA | NUMERICAL APPROXIMATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | NAVIER-STOKES EQUATIONS | ENGINEERING, MULTIDISCIPLINARY | ERROR ANALYSIS | FINITE-ELEMENT APPROXIMATIONS | CAHN-HILLIARD EQUATIONS | GLOBAL WEAK SOLUTIONS | Ferroalloys | Analysis | Mathematics - Numerical Analysis

Journal Article

Numerische Mathematik, ISSN 0029-599X, 7/2019, Volume 142, Issue 3, pp. 749 - 786

We introduce a discontinuous Galerkin method for the mixed formulation of the elasticity eigenproblem with reduced symmetry. The analysis of the resulting...

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 74B10 | 65N30 | MATHEMATICS, APPLIED

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 74B10 | 65N30 | MATHEMATICS, APPLIED

Journal Article

Advances in Applied Mathematics, ISSN 0196-8858, 09/2016, Volume 80, pp. 1 - 23

In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized...

Holonomic ansatz | Finite element method | Holonomic function | Inverse inequality | Zeilberger's algorithm | Symbolic determinant evaluation | 68W30 | 65N12 | 15A45 | MSC primary 33F10 | 05A20 | 65F15 | 15A15 | secondary 65N30 | MATHEMATICS, APPLIED | FINITE-ELEMENTS | COMPUTER ALGEBRA | PROOF | Analysis | Research institutes

Holonomic ansatz | Finite element method | Holonomic function | Inverse inequality | Zeilberger's algorithm | Symbolic determinant evaluation | 68W30 | 65N12 | 15A45 | MSC primary 33F10 | 05A20 | 65F15 | 15A15 | secondary 65N30 | MATHEMATICS, APPLIED | FINITE-ELEMENTS | COMPUTER ALGEBRA | PROOF | Analysis | Research institutes

Journal Article

Georgian Mathematical Journal, ISSN 1072-947X, 09/2018, Volume 25, Issue 3, pp. 337 - 348

In the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator...

65M15 | 65N12 | quasi-linear evolution equation | operator splitting | 65N22 | Decomposition scheme | MATHEMATICS

65M15 | 65N12 | quasi-linear evolution equation | operator splitting | 65N22 | Decomposition scheme | MATHEMATICS

Journal Article

Calcolo, ISSN 1126-5434, 2018, Volume 55, Issue 1, pp. 1 - 23

Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for...

Inverse inequality | 65N12 | Interpolation error estimate | Numerical Analysis | Virtual elements | Mathematics | Theory of Computation | Norm equivalence | 65N30 | MATHEMATICS | MATHEMATICS, APPLIED | 2ND-ORDER ELLIPTIC PROBLEMS | SPACES | DISCONTINUOUS GALERKIN METHOD

Inverse inequality | 65N12 | Interpolation error estimate | Numerical Analysis | Virtual elements | Mathematics | Theory of Computation | Norm equivalence | 65N30 | MATHEMATICS | MATHEMATICS, APPLIED | 2ND-ORDER ELLIPTIC PROBLEMS | SPACES | DISCONTINUOUS GALERKIN METHOD

Journal Article

Numerische Mathematik, ISSN 0029-599X, 3/2018, Volume 138, Issue 3, pp. 581 - 613

In the present work, we analyze the hp version of virtual element methods for the 2D Poisson problem. We prove exponential convergence of the energy error...

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 65N50 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | PIECEWISE ANALYTIC DATA | MATHEMATICS, APPLIED | WEIGHTED SOBOLEV SPACES | DISCONTINUOUS GALERKIN | 2ND-ORDER ELLIPTIC PROBLEMS | POLYGONAL MESHES | BOUNDARY-VALUE-PROBLEMS | POLYHEDRAL MESHES | STOKES PROBLEM | DIFFUSION-PROBLEMS | FRACTURE NETWORK SIMULATIONS

65N15 | Mathematical Methods in Physics | 65N12 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | 65N50 | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N30 | PIECEWISE ANALYTIC DATA | MATHEMATICS, APPLIED | WEIGHTED SOBOLEV SPACES | DISCONTINUOUS GALERKIN | 2ND-ORDER ELLIPTIC PROBLEMS | POLYGONAL MESHES | BOUNDARY-VALUE-PROBLEMS | POLYHEDRAL MESHES | STOKES PROBLEM | DIFFUSION-PROBLEMS | FRACTURE NETWORK SIMULATIONS

Journal Article