A \(hk\) mortar spectral element method for the \(p\)-Laplacian equation. (English) Zbl 1434.65298
Summary: We present a constrained-vertex variant of the mortar spectral element method to solve the \(p\)-Laplacian equation. To show reliability of the method, first we investigate convergence rate of \(h\)-version and \(k\)-version for a sufficiently smooth solution. Then for solutions with limited regularity, we use a \(hk\)-strategy in which the size of elements is geometrically reduced towards the singularity while the polynomial degree is linearly reduced. In fact, we numerically study convergence rate of the \(hk\) mortar spectral element method in the \(L^2\)-norm and \(H^1\)-norm using geometrical meshes with elements of non-uniform polynomial degree. In this regard, we consider several benchmarks with various choices of the parameter \(p\) as well as geometric grading factors. To deal with possible degeneracy in the \(p\)-Laplacian operator, we add artificial diffusion to the original equation and then study the effects of this extra diffusion on convergence rate. To find the solution of highly nonlinear system arising from discretization we use an optimization technique based on the trust region method in which an analytical Jacobian matrix is utilized.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 35J60 | Nonlinear elliptic equations |
Keywords:
mortar spectral element method; \(hp\) method; exponential convergence rate; \(p\)-Laplacian equation; nonlinear degenerate elliptic equations; torsional creep problemReferences:
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