fixed typos README.md (#95)

fixed typos

models/fenics-cookies-problem/README.md

@@ -42,7 +42,7 @@ Config          | Type      | Default      | Description
 BasisDegree     | Integer   | 1         | The degree of the piecewise polynomial FE approximation
 advection       | Integer   | 0         | A flag to add an an advection field to the problem. If set to 1 an advection term with advection field (see below) is added to the PDE problem.
 quad_degree     | Integer   | 8         | The quadrature degree used to evaluate integrals in the matrix assembly.
-diffzero        | Float     | 1.0       | Defines the background diffusion field $$a_0$$ (see below)
+diffzero        | Float     | 1.0       | Defines the background diffusion field $a_0$ (see below)
 directsolver    | Integer   | 1         | Uses a direct LU solve for linear system. Set to 0 to use GM-RES.
 pc              | String    | 'none'    | Preconditioning for the GM-RES solver, can be either 'ILU' or 'JACOBI'
 tol             | Float     | 1e-4      | Relative tolerance for the GM-RES solver.
@@ -68,7 +68,7 @@ Config          | Type      | Default      | Description
 BasisDegree     | Integer   | 1         | The degree of the piecewise polynomial FE approximation
 advection       | Integer   | 0         | A flag to add an an advection field to the problem. If set to 1 an advection term with advection field (see below) is added to the PDE problem.
 quad_degree     | Integer   | 8         | The quadrature degree used to evaluate integrals in the matrix assembly.
-diffzero        | Float     | 1.0       | Defines the background diffusion field $$a_0$$ (see below)
+diffzero        | Float     | 1.0       | Defines the background diffusion field $a_0$ (see below)
 letol           | Float     | 1e-4     | Local timestepping error tolerance for a simple implementation of TR-AB2 timestepping.
 T               | Float      | 10.0     | Final time for timestepping approximation. The QoI is evaluated and returned for time T.
 
@@ -87,7 +87,7 @@ None            |
 
 ![cookies-problem](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/models/fenics-cookies-problem/cookies_domain.png "geometry of the cookies problem")
 
-In its elliptic variant, the model implements the version of the cookies problem in [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3), see also e.g. [[Ballani et al.,2015]](https://doi.org/10.1137/140960980), [[Kressner et al., 2011]](https://doi.org/10.1137/100799010) for slightly different versions. With reference to the computational domain $$D=[0,1]^2$$ in the figure above, the cookies model consists in the thermal diffusion problem below, where $$\mathbf{y}$$ are the uncertain parameters discussed in the following and $$\mathrm{x}$$ are physical coordinates.  
+In its elliptic variant, the model implements the version of the cookies problem in [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3), see also e.g. [[Ballani et al.,2015]](https://doi.org/10.1137/140960980), [[Kressner et al., 2011]](https://doi.org/10.1137/100799010) for slightly different versions. With reference to the computational domain $D=[0,1]^2$ in the figure above, the cookies model consists in the thermal diffusion problem below, where $\mathbf{y}$ are the uncertain parameters discussed in the following and $\mathrm{x}$ are physical coordinates.  
 
 $$-\mathrm{div}\Big[ a(\mathbf{x},\mathbf{y}) \nabla u(\mathbf{x},\mathbf{y}) \Big] = f(\mathrm{x}), \quad \mathbf{x}\in D$$
 
@@ -98,7 +98,7 @@ $$f(\mathrm{x}) = \begin{cases}
 0 &\text{otherwise} 
 \end{cases}$$
 
-where $$F$$ is the square $$[0.4, 0.6]^2$$. The 8 subdomains with uncertain diffusion coefficient (the cookies) are circles with radius 0.13 and the following center coordinates:
+where $F$ is the square $[0.4, 0.6]^2$. The 8 subdomains with uncertain diffusion coefficient (the cookies) are circles with radius $0.13$ and the following center coordinates:
 
 cookie | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   |
 --     | --  | --  | --  | --  | --  | --  | --  | --  |
@@ -110,23 +110,23 @@ The uncertain diffusion coefficient is defined as
 $$a = a_0 + \sum_{n=1}^8 y_n \chi_n(\mathrm{x})$$ 
 
 
-where $$a_0=1$$ by default (can be changed by the user), $$y_n>-1$$ and 
+where $a_0=1$ by default (can be changed by the user), $y_n > -1$ and 
 
 $$\chi_n(\mathrm{x}) = \begin{cases} 1 &\text{inside the n-th cookie} \\ 0 &\text{otherwise} \end{cases}$$
 
 
-The output of the simulation is the integral of the solution over $$F$$, i.e. $$\Psi = \int_F u(\mathrm{x}) d \mathrm{x}$$
+The output of the simulation is the integral of the solution over $F$, i.e. $\Psi = \int_F u(\mathrm{x}) d \mathrm{x}$
 
 An advection term can be added to the equation, by suitably setting the config options. In this case, the equation becomes
 
 $$-\mathrm{div}\Big[ a(\mathbf{x},\mathbf{y}) \nabla u(\mathbf{x},\mathbf{y}) \Big] + \mathbf{w}(\mathbf{x}) \cdot \nabla u(\mathbf{x},\mathbf{y}) = f(\mathrm{x}), \quad \mathbf{x}\in D$$
 
-with $$\mathbf{w}(\mathbf{x})=  [4(x_2-0.5)(1-4(x_1-0.5)^2), \,\, -4(x_1-0.5)*(1-4(x_2-0.5)^2)].$$
+with $\mathbf{w}(\mathbf{x})=  [4(x_2-0.5)(1-4(x_1-0.5)^2), \,\, -4(x_1-0.5)*(1-4(x_2-0.5)^2)].$
 
-The PDE is solved by a classical Finite Eement Method (using the legacy version of [FEniCs](https://fenicsproject.org/) via the Python interface), with standard Lagrangian polynomial bases of degree $$p$$ ove quadrilateral meshes (FEM degree configurable by the user). The meshes contain $$100\times K$$ elements per direction. The corresponding linear system can be solvers either directly (LU solver) or iteratively by GMRES method, with three choices of preconditioning (none, ILU, Jacobi), up to the users. The relative tolerance of the iterative method can also be set by the user.
+The PDE is solved by a classical Finite Eement Method (using the legacy version of [FEniCs](https://fenicsproject.org/) via the Python interface), with standard Lagrangian polynomial bases of degree $p$ ove quadrilateral meshes (FEM degree configurable by the user). The meshes contain $100\times K$ elements per direction. The corresponding linear system can be solvers either directly (LU solver) or iteratively by GMRES method, with three choices of preconditioning (none, ILU, Jacobi), up to the users. The relative tolerance of the iterative method can also be set by the user.
 
 A parabolic (time-dependent) variant of the same problem is provided, that can be selected by using the forwardparabolic model introduced above. In this case, the equation becomes
 
 $$\displaystyle \frac{\partial u(\mathbf{x},t,\mathbf{y})}{\partial t} -\mathrm{div}\Big[ a(\mathbf{x},\mathbf{y}) \nabla u(\mathbf{x},\mathbf{y}) \Big] + \mathbf{w}(\mathbf{x}) \cdot \nabla u(\mathbf{x},t,\mathbf{y}) = f(\mathrm{x}), \quad \mathbf{x}\in D, t \in (0,T]$$
 
-with $$T$$ specified by the user and initial condition $$u=0$$. Like in the elliptic case, the advection term is null by default. All config options of the elliptic variant can be used here. Concerning time-stepping, the time-advancing scheme is adaptive with local error control, specifically an implementation of the TR-AB2 scheme, see, e.g., [[Iserles, 2008]](https://doi.org/10.1017/cbo9780511995569.009). 
+with $T$ specified by the user and initial condition $u=0$. Like in the elliptic case, the advection term is null by default. All config options of the elliptic variant can be used here. Concerning time-stepping, the time-advancing scheme is adaptive with local error control, specifically an implementation of the TR-AB2 scheme, see, e.g., [[Iserles, 2008]](https://doi.org/10.1017/cbo9780511995569.009).