Titlebook: An Introduction to Element-Based Galerkin Methods on Tensor-Product Bases; Analysis, Algorithms Francis X. Giraldo Textbook 2020 The Editor

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978-3-030-55071-4The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl

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Overview of Galerkin Methodsthe choices that we have at our disposal. We can categorize the possible methods as follows: .Generally speaking, the most widely used differential form method is the finite difference method while the most widely used integral form method is the Galerkin method (e.g., finite elements).

incontinence

Numerical Integration in One Dimensionus . and . are element and trace integrals, respectively. By element integrals we mean either area or volume integrals in 2D and 3D, respectively. By trace integrals we mean integrals along the boundary of the element which could be line or surface area integrals in 2D and 3D, respectively.

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1D Continuous Galerkin Methods for Elliptic Equationsonservation laws for both CG and DG. However, these types of equations are entirely hyperbolic (first order equations in these cases). In this chapter we learn how to use the CG method to discretize second order equations that are elliptic.

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Interpolation in Multiple Dimensionso and three dimensions. In one dimension, there is no room to choose the shape of the domain. That is, in the domain .?∈?[?1, +1] we are constrained to line elements. However, in two dimensions this door is flung wide open and we are now free to choose all sorts of polygons as the basic building blocks of our interpolation.

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2D Continuous Galerkin Methods for Hyperbolic Equationsly. In Ch. . we introduced the extension of the CG method to two dimensions by describing its implementation for elliptic partial differential equations (PDEs). In this chapter we extend the CG method for the application of hyperbolic equations in two dimensions. We also discuss the addition of diffusion operators.

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