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Lecture

Finite Difference Methods: Linear Systems and Band Matrices

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 application of finite difference methods to solve partial differential equations, focusing on the theoretical aspects and practical exercises. The instructor explains the aquosination of Laplace problems, dimensionality, and the use of band matrices in linear systems.

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Instructor

Annalisa Buffa

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Related concepts (32)

Band matrix

In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. Formally, consider an n×n matrix A=(ai,j ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k1 and k2: then the quantities k1 and k2 are called the and , respectively. The of the matrix is the maximum of k1 and k2; in other words, it is the number k such that if .

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.

Sparse matrix

In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.

Skyline matrix

In scientific computing, skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces the storage requirement of a matrix more than banded storage. In banded storage, all entries within a fixed distance from the diagonal (called half-bandwidth) are stored. In column-oriented skyline storage, only the entries from the first nonzero entry to the last nonzero entry in each column are stored.

Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

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