In the present paper, both symmetric and nonsymmetric interior penalty discontinuous \(hp\)-Galerkin methods are applied to a class of quasi-linear elliptic problems which are of nonmonotone type.
The first section is an introduction in nature.
The second section is devoted to notation, definitions and preliminaries.
The third section is related to discontinuous Galerkin methods for linear nonselfadjoint elliptic problems of type:
\[ \begin{cases} -\nabla\cdot(a(x)\nabla u)+\vec{b}(x)\cdot\nabla u+a_0(x)u=f(x)&\text{in } \Omega\\ u=g&\text{on } \partial\Omega \end{cases}\tag{1} \]
A priori error estimates are derived in the broken \(H^1\)-norm, which are optimal in \(h\) and suboptimal in \(p.\)
The fourth section is devoted to symmetric interior penalty discontinuous Galerkin (SPIG) and nonsymmetric interior penalty discontinuous Galerkin (NIPG) methods for quasi-linear elliptic problems. Using Brouwer’s fixed theorem, existence of a discrete solution is proved. Then a priori error estimates are derived in the broken \(H^1\)-norm, which are optimal in \(h\) and suboptimal in \(p.\) Further, an a priori error estimate in the \(L^2\)-norm is established on regular meshes for elliptic problems with piecewise polynomial or zero Dirichlet boundary datum.
In order to illustrate the theoretical results obtained in the present material, some numerical experiments are provided within the fifth section. For this, one considers the nonlinear elliptic problem:
\[ \begin{cases} -\nabla\cdot((1+u)\nabla u)=f&\text{ in }\Omega\\ u=0&\text{ on } \partial\Omega, \end{cases} \]
where \(\Omega=(0,1)\times(0,1)\) and \(f\) is taken in such way that the exact solution is \(u=x(1-x)y(1-y).\)
The last section presents a summary and some extensions.