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Numerical studies of groundwater flow problems with a singularity

Hokr, Milan, Balvín, Aleš (2017)

Programs and Algorithms of Numerical Mathematics

The paper studies mesh dependent numerical solution of groundwater problems with singularities, caused by boreholes represented as points, instead of a real radius. We show on examples, that the numerical solution of the borehole pumping problem with point source (singularity) can be related to the exact solution of a regular problem with adapted geometry of a finite borehole radius. The radius providing the fit is roughly proportional to the mesh step. Next we define a problem of fracture-rock...

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Runchang Lin, Zhimin Zhang (2009)

Applications of Mathematics

Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.

Numerical study of two sparse AMG-methods

Janne Martikainen (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

Numerical Study of Two Sparse AMG-methods

Janne Martikainen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

Numerical treatment of 3-dimensional potential problem

Vladimír Drápalík, Vladimír Janovský (1988)

Aplikace matematiky

Assuming an incident wave to be a field source, we calculate the field potential in a neighborhood of an inhomogeneous body. This problem which has been formulated in 𝐑 3 can be reduced to a bounded domain. Namely, a boundary condition for the potential is formulated on a sphere. Then the potential satisfies a well posed boundary value problem in a ball containing the body. A numerical approximation is suggested and its convergence is analyzed.

Object oriented design philosophy for scientific computing

Philippe R. B. Devloo, Gustavo C. Longhin (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

Object oriented design philosophy for scientific computing

Philippe R.B. Devloo, Gustavo C. Longhin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

Observations Regarding Algorithms Required for Robust CFD Codes

F. T. Johnson, D. S. Kamenetskiy, R. G. Melvin, V. Venkatakrishnan, L. B. Wigton, D. P. Young, S. R. Allmaras, J. E. Bussoletti, C. L. Hilmes (2011)

Mathematical Modelling of Natural Phenomena

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined the wind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFD has had its most favorable impact on the aerodynamic design of the high-speed cruise configuration of a transport. This success has raised expectations among aerodynamicists that the applicability of CFD can be extended to the full flight envelope. However, the complex nature...

On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions

Jean-Luc Guermond, Luigi Quartapelle, Jiang Zhu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element...

On a certain two-sided symmetric condition in magnetic field analysis and computations

František Melkes, Alexander Ženíšek (1997)

Applications of Mathematics

A special two-sided condition for the incremental magnetic reluctivity is introduced which guarantees the unique existence of both the weak and the approximate solutions of the nonlinear stationary magnetic field distributed on a region composed of different media, as well as a certain estimate of the error between the two solutions. The condition, being discussed from the physical as well as the mathematical point of view, can be easily verified and is fulfilled for various magnetic reluctivity...

On a method of two-sided eigenvalue estimates for elliptic equations of the form A u - λ B u = 0

Karel Rektorys, Zdeněk Vospěl (1981)

Aplikace matematiky

The Collatz method of twosided eigenvalue estimates was extended by K. Rektorys in his monography Variational Methods to the case of differential equations of the form A u - λ B u = 0 with elliptic operators. This method requires to solve, successively, certain boundary value problems. In the case of partial differential equations, these problems are to be solved approximately, as a rule, and this is the source of further errors. In the work, it is shown how to estimate these additional errors, or how to avoid...

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