doc: fix typos in tutorials (#1131)

docs/src/literate-tutorials/dg_heat_equation.jl

@@ -113,7 +113,7 @@
 #      \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla}] (\delta u) \,\mathrm{d}\Omega - \int_\Gamma \llbracket u \rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} + \llbracket \delta u \rrbracket  \cdot \{\boldsymbol{\nabla} (u)\}  \,\mathrm{d}\Gamma + \int_\Gamma \frac{\eta}{h_e} \llbracket u\rrbracket  \cdot \llbracket \delta u\rrbracket   \,\mathrm{d}\Gamma = \int_\Omega \delta u \,\mathrm{d}\Omega,\\
 #     ```
 # Since $\partial \Omega$ is constrained with both Dirichlet and Neumann boundary conditions the term $\int_{\partial \Omega} [\boldsymbol{\nabla} (u)] \cdot \boldsymbol{n} \delta u \,\mathrm{d} \Omega$ can be expressed as an integral over $\partial \Omega_N$, where $\partial \Omega_N$ is the boundaries with only prescribed Neumann boundary condition,
-# The resulting weak form is given given as follows: Find $u \in \mathbb{U}$ such that
+# The resulting weak form is given as follows: Find $u \in \mathbb{U}$ such that
 # ```math
 #  \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega - \int_{\Gamma^0} \llbracket u\rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} + \llbracket \delta u\rrbracket  \cdot \{\boldsymbol{\nabla} (u)\}  \,\mathrm{d}\Gamma^0 + \int_{\Gamma^0} \frac{\eta}{h_e} \llbracket u\rrbracket \cdot \llbracket \delta u\rrbracket   \,\mathrm{d}\Gamma^0 = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_{\partial \Omega_N} ([\boldsymbol{\nabla} (u)] \cdot \boldsymbol{n}) \delta u \,\mathrm{d} \partial \Omega_N,\\
 # ```

docs/src/literate-tutorials/heat_equation.jl

@@ -28,7 +28,7 @@
 # u(\textbf{x}) = 0 \quad \textbf{x} \in \partial \Omega,
 # ```
 # where $\partial \Omega$ denotes the boundary of $\Omega$.
-# The resulting weak form is given given as follows: Find ``u \in \mathbb{U}`` such that
+# The resulting weak form is given as follows: Find ``u \in \mathbb{U}`` such that
 # ```math
 # \int_{\Omega} \nabla \delta u \cdot \nabla u \ d\Omega = \int_{\Omega} \delta u \ d\Omega \quad \forall \delta u \in \mathbb{T},
 # ```
@@ -118,7 +118,7 @@ close!(ch)
 #
 # #### Element assembly
 # We define the function `assemble_element!` (see below) which computes the contribution for
-# an element. The function takes pre-allocated `ke` and `fe` (they are allocated once and
+# an element. The function takes pre-allocated `Ke` and `fe` (they are allocated once and
 # then reused for all elements) so we first need to make sure that they are all zeroes at
 # the start of the function by using `fill!`. Then we loop over all the quadrature points,
 # and for each quadrature point we loop over all the (local) shape functions. We need the

docs/src/literate-tutorials/linear_elasticity.jl

@@ -43,7 +43,7 @@
 # denotes the symmetric part.
 #
 # ### Weak form
-# The resulting weak form is given given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that
+# The resulting weak form is given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that
 # ```math
 # \int_\Omega
 #   \mathrm{grad}(\delta \boldsymbol{u}) : \boldsymbol{\sigma}