Numerical solution of heat conduction problems using orthogonal collocation on finite elements (Q2833233)

The paper is concerned with the numerical solution of the nonlinear parabolic equation NEWLINE\[NEWLINE\frac{\partial y}{ \partial t}= \epsilon \frac{\partial^2 y}{\partial x^2} - \beta \left(x\right) \frac{\partial y}{\partial x}- f \left(y\right), \quad \left(x, t \right) \in \left( 0,1 \right) \times \left(0,1\right), NEWLINE\]NEWLINE equipped with initial conditions and Robin boundary conditions. The problem is solved using an orthogonal collocation method with Lagrange interpolating polynomials. The zeros of the shifted Jacobi polynomials are chosen as collocation points.NEWLINENEWLINEError estimates are recalled for Lagrange polynomial interpolation and convergence of the method is proved. Numerical examples are presented for both linear and nonlinear parabolic problems and various choices of the parameter \(\epsilon\).