By Bo-nan Jiang

ISBN-10: 3642083676

ISBN-13: 9783642083679

ISBN-10: 3662037408

ISBN-13: 9783662037409

This is the 1st booklet dedicated to the least-squares finite aspect technique (LSFEM), that is an easy, effective and powerful method for the numerical answer of partial differential equations. The e-book demonstrates that the LSFEM can clear up a wide diversity of difficulties in fluid dynamics and electromagnetics with just one mathematical/computational formula. The e-book indicates that more often than not followed distinct remedies in computational fluid dynamics and computational electromagnetics, similar to upwinding, numerical dissipation, staggered grid, non-equal-order components, operator splitting and preconditioning, aspect components, vector power, etc, are unnecessary.

This booklet introduces the elemental thought of the least-squares technique for first-order PDE structures, rather the div-curl method and the div-curl-grad process. it's utilized to the examine of permissible boundary stipulations for the incompressible Navier--Stokes equations, to teach that the divergence equations within the Maxwell equations should not redundant, and to derive an identical second-order models of the Navier--Stokes equations and the Maxwell equations. This e-book covers diversified functions akin 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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**Additional resources for The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics**

**Example text**

The definition of C1 element will be given in Sect. ) Mixed Galerkin methods remain a delicate class of finite element methods in that it is not easy to choose the approximation spaces and verify the satisfaction of the LBB condition. 20) iteratively in an efficient way for large-scale problems. 1 The Least-Squares Formulation Before introducing the least-squares formulation let us recall a few simple concepts. Assume that V is a linear space. We say that L is a linear form 38 3. , for all = (3L(v) + OL(w).

Xn-l < Xn = 1 to divide the interval [0, 1J into n subintervals or elements ej = (Xj-l, Xj), j = 1,2, ... , n, of length hj = Xj - Xj-I. and set h = max hj. The quantity h is then a measure of how fine the partition is. x Fig. 1. Example of a function Uh E Vh We now construct a trial function Uh such that Uh is linear on each element ej, and Uh is continuous on [0,1] and satisfies the boundary condition Uh(O) = 0, as illustrated in Fig. 1. These functions constitute the subspace Vh , and Vh C V.

Jiang, The Least-Squares Finite Element Method © Springer-Verlag Berlin Heidelberg 1998 32 3. 2 The Rayleigh-Ritz Method We have pointed out in Chap. 2 that the Galerkin method is based on the basic lemma of variational principles. , Washizu 1975). It is well-known that for diffusion and elasticity problems, the Galerkin method results in the same finite element scheme as in the Rayleigh-Ritz method which is based on the principle of minimum potential energy. 1) we first introduce the function space H = {u E Hl(O, 1) : u(O) = u(l) = O}.

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