# PDF Numerical Solution of Partial Differential Equations on Parallel Computers

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MIT Numerical Methods for PDE Lecture 3: Finite Difference for 2D Poisson's equation

Today, many important simulation problems can not be solved by one single computer, but calls for parallel computing. Enter your Postcode or Suburb to view availability and delivery times.

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See Terms for more information. Contact 07 online qbd. We introduce the derivative of functions using discrete Fourier transforms and use it to solve the 1D and 2D acoustic wave equation. The necessity to simulate waves in limited areas leads us to the definition of Chebyshev polynomials and their uses as basis functions for function interpolation.

We develop the concept of differentiation matrices and discuss a solution scheme for the elastic wave equation using Chebyshev polynomials. Week 06 The Linear Finite-Element Method - Static Elasticity -We introduce the concept of finite elements and develop the weak form of the wave equation.

## Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python

We discuss the Galerkin principle and derive a finite-element algorithm for the static elasticity problem based upon linear basis functions. We also discuss how to implement boundary conditions. The finite-difference based relaxation method is derived for the same equation and the solution compared to the finite-element algorithm. Week 07 The Linear Finite-Element Method - Dynamic Elasticity -We extend the finite-element solution to the elastic wave equation and compare the solution scheme to the finite-difference method.

To allow direct comparison we formulate the finite-difference solution in matrix-vector form and demonstrate the similarity of the linear finite-element method and the finite-difference approach. We introduce the concept of h-adaptivity, the space-dependence of the element size for heterogeneous media.

## CRSC Research Areas

Week 08 The Spectral-Element Method - Lagrange Interpolation, Numerical Integration -We introduce the fundamentals of the spectral-element method developing a solution scheme for the 1D elastic wave equation. Lagrange polynomials are discussed as the basis functions of choice.

The concept of Gauss-Lobatto-Legendre numerical integration is introduced and shown that it leads to a diagonal mass matrix making its inversion trivial. We show how to calculate the required derivatives of the Lagrange polynomials making use of Legendre polynomials. We show how to perform the assembly step leading to the final solution system for the elastic wave equation.

### Parallel Multilevel Methods for Elliptic Partial Differential Equations

We demonstrate the numerical solution for homogenous and heterogeneous media. Taught by Heiner Igel. Tags numerical methods. Browse More Coursera Articles. Stanford University Machine Learning via Coursera.