17,260 research outputs found
Multigrid elliptic equation solver with adaptive mesh refinement
In this paper we describe in detail the computational algorithm used by our
parallel multigrid elliptic equation solver with adaptive mesh refinement. Our
code uses truncation error estimates to adaptively refine the grid as part of
the solution process. The presentation includes a discussion of the orders of
accuracy that we use for prolongation and restriction operators to ensure
second order accurate results and to minimize computational work. Code tests
are presented that confirm the overall second order accuracy and demonstrate
the savings in computational resources provided by adaptive mesh refinement.Comment: 12 pages, 9 figures, Modified in response to reviewer suggestions,
added figure, added references. Accepted for publication in J. Comp. Phy
VAGO method for the solution of elliptic second-order boundary value problems
Mathematical physics problems are often formulated using differential
oprators of vector analysis - invariant operators of first order, namely,
divergence, gradient and rotor operators. In approximate solution of such
problems it is natural to employ similar operator formulations for grid
problems, too. The VAGO (Vector Analysis Grid Operators) method is based on
such a methodology. In this paper the vector analysis difference operators are
constructed using the Delaunay triangulation and the Voronoi diagrams. Further
the VAGO method is used to solve approximately boundary value problems for the
general elliptic equation of second order. In the convection-diffusion-reaction
equation the diffusion coefficient is a symmetric tensor of second order
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
A posteriori error control for fully discrete Crank–Nicolson schemes
We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank--Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, which is introduced in the present paper
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
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