By Bo-nan Jiang

This is the 1st ebook dedicated to the least-squares finite aspect approach (LSFEM), that is an easy, effective and powerful method for the numerical resolution of partial differential equations. The publication demonstrates that the LSFEM can clear up a extensive variety of difficulties in fluid dynamics and electromagnetics with just one mathematical/computational formula. The ebook exhibits that more often than not followed particular remedies in computational fluid dynamics and computational electromagnetics, corresponding to upwinding, numerical dissipation, staggered grid, non-equal-order components, operator splitting and preconditioning, part components, vector strength, and so forth, are unnecessary.
This booklet introduces the elemental concept of the least-squares approach for first-order PDE platforms, really the div-curl process and the div-curl-grad process. it really is utilized to the examine of permissible boundary stipulations for the incompressible Navier--Stokes equations, to teach that the divergence equations within the Maxwell equations are usually not redundant, and to derive an identical second-order models of the Navier--Stokes equations and the Maxwell equations. This ebook covers different functions equivalent to incompressible viscous flows, rotational inviscid flows, low- or high-Mach-number compressible flows, two-fluid flows, convective flows, and scattering waves.

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1. We obtained the rate of convergence of the method by calculating the norm of the error for each h, plotting log II error II versus log h, and calculating the slope of this line. 51). 6 g: ... og(h) Fig. 1. 1. 5 ± order f order m 1 2 1 2 Ilepll lIe,,1I 44 3. 5 Concluding Remarks For the solution of diffusion-type differential equations, both the mixed Galerkin method and the least-squares method are based on the first-order system generated by introducing the dual variable p (the flux) as an additional unknown.

Let E = u- Ih U be the interpolation error function and consider an arbitrary element ej with points Xj-l :::; x :::; Xj in the mesh. We assume that U E H2(0, 1), and let luI2 = {Jorl u"(x)2dx }1/2 . This is the semi-norm of H2(0, 1) introduced in Sect. 2. Since Ihu is the interpolant of u, the error E vanishes at both endpoints Xj-l and Xj' By virtue of the Rolle theorem, there exists at least one point x between Xj-l and Xj at which E'(x) = 0. Then for any x, 20 2. First-Order Scalar Equation in One Dimension l E'(x) = x E"(t)dt.

Prenter (1975), Oden and Demkowicz (1996). A linear space U equipped with a norm is called a normed space and is denoted by the pair {U, II· II,,}. We use this notation to emphasize that the particular norm I . II" is associated with the space U. • , of elements in U which satisfies the following property: For any E > 0 there exists a natural number N such that the distance lIui - Ujll" < E if i,j > N. Further, a sequence in U is said to converge to an element U if the distance Ilui - ull" -+ 0 as i -+ 00.

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