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Lecture

Finite Element Method: Galerkin Method

In course

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions. ...

Description

This lecture covers the Finite Element Method, focusing on the Galerkin method for solving non-homogeneous Dirichlet problems. It explains the concept of conformal triangulation, affine transformations, and the importance of triangular meshes in the method.

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Instructor

Annalisa Buffa

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Related concepts (32)

Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments.

Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point.

Affine group

In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line. Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars the real or complex field, then the affine group is a Lie group.

Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (Given a line L and a point P not on L, there is exactly one line parallel to L that passes through P.) is fundamental in affine geometry.

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