One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems
We consider the analysis of a one-parameter family of $hp$--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form $u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u)$ on a bounded open set $\om\in\re^d$, subject to mixed D...
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| Format: | Report |
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Unspecified
2004
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| _version_ | 1797068149261074432 |
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| author | Lasis, A Suli, E |
| author_facet | Lasis, A Suli, E |
| author_sort | Lasis, A |
| collection | OXFORD |
| description | We consider the analysis of a one-parameter family of $hp$--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form $u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u)$ on a bounded open set $\om\in\re^d$, subject to mixed Dirichlet and Neumann boundary conditions on $\pr\om$. It is assumed that $a$ is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and $f$ is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform $hp$--meshes, if $u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om))$ with $k\geq 3\frac{1}{2}$, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken $H^1$ norm, is proved to be the same as in the linear case: $\mathscr{O}(h^{s-1}/p^{k-3/2})$ with $1\leq s\leq\min\set{p+1,k}$. |
| first_indexed | 2024-03-06T22:06:32Z |
| format | Report |
| id | oxford-uuid:505acf88-8e96-4b77-911b-87e44d44de0c |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:06:32Z |
| publishDate | 2004 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:505acf88-8e96-4b77-911b-87e44d44de0c2022-03-26T16:13:00ZOne-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problemsReporthttp://purl.org/coar/resource_type/c_93fcuuid:505acf88-8e96-4b77-911b-87e44d44de0cMathematical Institute - ePrintsUnspecified2004Lasis, ASuli, EWe consider the analysis of a one-parameter family of $hp$--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form $u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u)$ on a bounded open set $\om\in\re^d$, subject to mixed Dirichlet and Neumann boundary conditions on $\pr\om$. It is assumed that $a$ is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and $f$ is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform $hp$--meshes, if $u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om))$ with $k\geq 3\frac{1}{2}$, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken $H^1$ norm, is proved to be the same as in the linear case: $\mathscr{O}(h^{s-1}/p^{k-3/2})$ with $1\leq s\leq\min\set{p+1,k}$. |
| spellingShingle | Lasis, A Suli, E One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title | One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title_full | One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title_fullStr | One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title_full_unstemmed | One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title_short | One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems |
| title_sort | one parameter discontinuous galerkin finite element discretisation of quasilinear parabolic problems |
| work_keys_str_mv | AT lasisa oneparameterdiscontinuousgalerkinfiniteelementdiscretisationofquasilinearparabolicproblems AT sulie oneparameterdiscontinuousgalerkinfiniteelementdiscretisationofquasilinearparabolicproblems |