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Minimum residual discretization of a semilinear elliptic problem

Carlos García Vera, Norbert Heuer, Dirk Praetorius

Abstract

We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme.

Minimum residual discretization of a semilinear elliptic problem

Abstract

We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme.

Paper Structure

This paper contains 7 sections, 4 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction and model problem
  2. Minimum residual formulation
  3. Discretization
  4. Numerics
  5. Nonlinear systems
  6. Adaptivity
  7. Experiments

Key Result

Theorem 1

Suppose that pde has a solution. Then, any solution $u\in H^1_0(\Omega)$ of pde gives rise to a solution $\boldsymbol{u}=(u,{\widehat{\sigma}},q,r)$ of min with ${\widehat{\sigma}}=\mathop{\mathrm{tr_\mathit{n}}}\nolimits(\kappa\nabla u+q\beta)$, $q=\rho(u)$, and $r=\gamma(u)$. On the other hand, an

Figures (4)

  • Figure 1: Example 1. Errors and residuals, uniform refinements.
  • Figure 2: Example 2. Errors and residuals, uniform refinements.
  • Figure 3: Example 2. Errors and residuals, adaptive refinements.
  • Figure 4: Example 2. Exact solutions $u$, $q=\rho(u)$, $r=\gamma(u)$ (top) and their approximations (bottom) after $3$ adaptive refinements, $N=196$.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Corollary 5
  • proof