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Iterative techniques are a key methodology for the numerical solution of optimization problems in differential equations. In two practical application problems with different characteristics, this paper shows, how multigrid methods can be applied efficiently to this problem class. Problem formulations, solution approaches as well as numerical results are presented.

In this article we are interested in the derivation of efficient domain decomposition methods for the viscous primitive equations of the ocean. We consider the rotating 3d incompressible hydrostatic Navier-Stokes equations with free surface. Performing an asymptotic analysis of the system with respect to the Rossby number, we compute an approximated Dirichlet to Neumann operator and build an optimized Schwarz waveform relaxation algorithm. We establish the well-posedness of this algorithm and present some numerical results to illustrate the method.

We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretizations of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach., Comment: 25 pages

Residual distributions (RD) schemes are a class of of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [J. Comp. Phys. 172, 572 (2001)] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)]. In this work we extend LeVeque and Pelanti's results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an m-by-m matrix in each computational grid cell, where $m$ is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive., Comment: 46 pages, 14 figures

Mathematical models of many pattern formation problems in biology are reaction-diffusion systems. These systems are important for computer simulations of the patterns, parameter estimations as well as analysis of the biological properties. In order to solve reaction-diffusion systems efficiently, fast and stable numerical algorithms are essential for the pattern formation problems. In this paper, a fairly general reaction-diffusion system is considered. We propose a fully implicit discretization combined with a multigrid V-cycle solver for solving the reaction-diffusion system. Theorems about unconditional stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results are given for two reaction-diffusion systems that can be used for generating animal coat markings. We also show the comparison results of the multigrid algorithm with other numerical algorithms.