XLiFE++ website is now hosted here with up-to-date content. The current website, with potentially deprecated content, will be removed soon.

Example 7/18 - Discontinuous Galerkin method for 2D Dirichlet problem

Consider the Laplace problem with homogeneous Dirichlet condition:

$$\{\begin{array}{cc}-\mathrm{\Delta }u=f\hfill & \text{ in }\mathrm{\Omega }\hfill \\ u=0\hfill & \text{ on }\partial \mathrm{\Omega }\hfill \end{array}$$

Consider a discontinuous Galerkin space $V_h$, e.g. discontinuous $P^1$-Lagrange space and let introduce the IP (Interior Penalty) formulation:

$$|\begin{array}{c}\text{find }{u}_h\in V_h\text{ such that }\hfill \\ {\int \; <!--nolimits-->}_{\mathrm{\Omega }}{\nabla <!--FUNCTION\; APPLICATION-->}_h{u}_h.{\nabla <!--FUNCTION\; APPLICATION-->}_{h}v-{\int \; <!--nolimits-->}_{\Gamma }\{{\nabla <!--FUNCTION\; APPLICATION-->}_h{u}_h.n\}[v]-{\int \; <!--nolimits-->}_{\Gamma }[u_h]\{{\nabla <!--FUNCTION\; APPLICATION-->}_{h}v.n\}+{\int \; <!--nolimits-->}_{\Gamma }\mu [u_h][v]={\int \; <!--nolimits-->}_{\mathrm{\Omega }}fv\forall <!--FUNCTION\; APPLICATION-->v\in V_h.\hfill \end{array}$$

where $\mathrm{\Omega }=\bigcup <!--FUNCTION\; APPLICATION-->T_{\ell }$, $\Gamma =\bigcup <!--FUNCTION\; APPLICATION-->\partial T_{\ell }$ (the set of all sides of mesh elements) and, $S_{k,\ell }$ denoting the side shared by the elements $T_k$ and $T_{\ell }$:

$$\{v{\}}_{|S_{k\ell }}=\frac{1}{2}\left({\left(v_{|T_k}\right)}_{|S_{k\ell }}+{\left(v_{|T_{\ell }}\right)}_{|S_{k\ell }}\right)[v{]}_{|S_{k\ell }}=\left({\left(v_{|T_k}\right)}_{|S_{k\ell }}-{\left(v_{|T_{\ell }}\right)}_{|S_{k\ell }}\right).$$

For a non shared side $S_k$, we set

$$\{v{\}}_{|S_k}=[v{]}_{|S_k}={\left(v_{|T_k}\right)}_{|S_k}.$$

The operators $\{.\}$ and $[.]$ are implemented in XLiFE++ as mean(.) and jump(.) operators. The normal vector $n$ is chosen as the outward normal from $T_k$ on side $S_{k\ell }$; in practice outward from the "first" parent element of the side. $\mu$ is a penalty function usually chosen to be proportional to the inverse of the measure of the side $S_{k\ell }$. Note that the Dirichlet boundary condition is a natural condition in this formulation.

To deal with a such formulation, the following objects have to be constructed from a geometrical domain, say omega :

• a FE space specifying $L2$ conformity (discontinuous space) defined on omega

• the geometrical domain Gamma of sides of omega using the function sides(omega)

• the bilinear form related to IP formulation and involving mean(.) and jump(.) operators

The XLiFE++ implementation of this problem using discontinuous P1-Lagrange is the following:

The $L^2$ error is about $0.00317$ for 5400 dofs. Using Paraview with the Warp by scalar filter that produces elevation, the approximated field $u$ looks like:

---

---

It can be noticed that the field is discontinuous and major errors is located on the corners. NIPG formulation is an other penalty method corresponding to the bilinear form:

$${\int \; <!--nolimits-->}_{\mathrm{\Omega }}{\nabla <!--FUNCTION\; APPLICATION-->}_h{u}_h.{\nabla <!--FUNCTION\; APPLICATION-->}_{h}v-{\int \; <!--nolimits-->}_{\Gamma }\{{\nabla <!--FUNCTION\; APPLICATION-->}_h{u}_h.n\}[v]+{\int \; <!--nolimits-->}_{\Gamma }[u_h]\{{\nabla <!--FUNCTION\; APPLICATION-->}_{h}v.n\}+{\int \; <!--nolimits-->}_{\Gamma }\mu [u_h][v]$$

that is close to the IP formulation but non symmetric. For NIPG, the $L^2$ error is weaker (about $0.00141$) and the solution is less polluted at the corners.

• Tutorial
  • HTML, PDF
• Install
  • HTML, PDF
• Examples
  • HTML, PDF
• User Documentation
  • HTML, PDF
  • Older pdfs
• Doxygen (HTML)
  • Last release