# The Basics

## Contents

tutorials:basics

# 2. The Basics#

We learn the basic concepts to program in NGSolve, we start with an easy example and we explore the single steps to solve a PDE problem. Then we apply the same steps and concepts to a more complex problem.

basic topics

## 2.1. Finite Element Method in a Nutshell#

Problem: (Strong)
Let $$\Omega$$ be a domain in $$\mathbb{R}^n$$ .

Find $$u$$ such that

\begin{align*} -\Delta u &= f \quad \text{in } \Omega, \\ u &= u_D \quad \text{on } \Gamma_D,\\ n\cdot \nabla u &= g \quad \text{on } \Gamma_N, \end{align*}

With $$\Gamma_D \sqcup \Gamma_N = \partial \Omega$$

Multiply the equation by a test function $$v$$ and integrate by parts we get the weak formulation:

Problem: (Weak)
Find $$u \in H^1(\Omega)$$ such that $$u = u_D$$ on $$\Gamma_D$$ and

\begin{align*} \int_{\Omega} \nabla u \cdot \nabla v \, dx &= \int_{\Omega} f v \, dx - \int_{\Gamma_N} g v \, ds \quad \forall v \in H^1_{\Gamma_D}(\Omega), \end{align*}

Let $$\mathcal{T}$$ ba a triangulation of the space $$\Omega$$ the discrete weak formulation is given by:

Problem: (Discrete-Weak)
Find $$u_h \in V_{h}$$ such that $$u_h = u_D$$ on $$\Gamma_D$$ and

\begin{align*} \int_{\Omega} \nabla u_h \cdot \nabla v_h \, dx &= \int_{\Omega} f v_h \, dx - \int_{\Gamma_N} g v_h \, ds\quad \forall v_h \in V_{h,\Gamma_D}(\mathcal{T}), \end{align*}

Applying the Galerkin method: Suppose that on the triangulation $$\mathcal{T}$$ we the finite dimensional function space $$V_{h}(\mathcal{T})$$ has a basis $$\{\phi^i\}_{i=1}^{N}$$, and let

\begin{align*} v_h = \sum_{i=1}^{N} v^i \phi_i\quad \text{and} \quad u_h = \sum_{j=1}^{N} u^j \phi_j \end{align*}

Then, by substituting $$v_h$$ and $$u_h$$ in the discrete weak formulation we obtain the following linear system:

\begin{align*} \sum_{i,j} v^i u^j \overbrace{\int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i \, dx }^{A_{ij}}&= \sum_{i} v^i \overbrace{ \left(\int_{\Omega} f \phi_i \, dx - \int_{\Gamma_N} g \phi_i \, ds \right)}^{f_i} \end{align*}

Then the problem reduces to solve the linear system $$A u = f$$.