Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

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First two integrals in (3.59) can be transformed into a single integral along the whole boundary ∂ Ω . In addition, subintegral functions in the first two integral in (3.57) contain highest first derivatives with respect to u and v , and thus can be transformed into a single integral along the whole domain Ω , assuming that functions u and v are smooth enough: Z Ω ( k ∇ u · ∇ v − f v ) d x d y − Z ∂ Ω k ∂ u ∂ n v d s = 0 . (3.61)

Replacing of natural boundary conditions (3.54) into the linear integral gives: Z Ω ( k ∇ u · ∇ v − f v ) d x d y + Z ∂ Ω ˆ σ v d s = 0 ,

(3.62)

Which applies to all allowable weight functions v . If this solutio data is sufficiently smooth, the solution of equation (3.62) is also the solution of (3.54). Vice versa, every solution of (3.54) is automatically the solution of equation (3.62). An important question still remains about determining of the adequate class of allowable functions for problem (3.62). Let us observe that the integrals in (3.62) are defined if functions u and v and their partial derivatives are smooth enough for square integration with respect to Ω : Z Ω " ∂ v ∂ x 2 + ∂ v ∂ y 2 + v 2 # d x d y < ∞ . (3.63) Such class of functions will be denoted by H 1 ( Ω ) , where 1 refers to the „square integrability“ of the first derivative, and Ω is the domain of the defined functions. As in the case of one-dimensional problems, natural boundary conditions are the integral part of equation (3.62), and they appear in term R ∂ Ω 2 ˆ σ v d s . Essential boundary conditions are taken into account via allowable functions definition. For weight functions, we will select functions v from H 1 ( Ω ) which are equal to zero at ∂ Ω 1 , and their solution must be a function within H 1 ( Ω ) , for which u = ˆ u at ∂ Ω 1 . Variation formulation of the boundary condition is as follows: find a function u ∈ H 1 ( Ω ) such that u = ˆ u at ∂ Ω 1 and (3.63) holds for ∀ v ∈ H 1 ( Ω ) , such that v = 0 at ∂ Ω 1 . Thus, in addition to the aforementioned variation formulation properties (see text after (3.62), the following holds: • Constraint imposed by the variation formulation are weaker than the constraints that apply to equation (3.54). • The jump condition (3.52) does not require additional consideration in the case of variation formulation 3.4 Finite element interpolation This chapter represents a direct generalisation of the corresponding one-dimensional problem analysis chapter, but with significant differences in certain important details. Above all, in the case of the one-dimensional problems, the finite element mesh is defined simply be dividing a linear domain into linear subdomains, by introducing nodes in all discontinuities. For two-dimensional problems, discretisation is not that simple. Essentially, we still tend to represent the approximate solution u h and weight functions v h via polynomials defined on subdomains with simple geometry, for a given area Ω h , which is located in the plane x , y . Additionally, discretization should be general enough to model irregular domains, while containing elements which are sufficiently simple to calculate. For this purpose, the most favourable shapes are the triangle and the rectangle, Fig. 3.9